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MATHEMATICAL MONOGRAPHS. 

EDITED BY 

MANSFIELD MERRIMAN AND ROBERT S. WOODWARD. 






No. 4. 



HYPERBOLIC FUNCTIONS 



BY 



JAMES McMAHON, 

PROFESSOR OF MATHEMATICS IN CORNELL UNIVERSITY. 



FOURTH EDITION, ENLARGED. 
FIRST THOUSAND. 



NEW YORK: 

JOHN WILEY & SONS. 

LONDON: CHAPMAN & HALL, LIMITED. 

1906. 




COPYRIGHT, 1896, 

BY 
MANSFIELD MKRRIMAN AND ROBERT S. WOODWARD 

UNDER THE TITLE 

HIGHER MATHEMATICS. 

First Edition, September, 1896. 
Second Edition, January, 1898. 
Third Edition, August, 1900. 
Fourth Edition, January, 1906. 




UOBKRT DRUMMOND, PWTNTPT?, 



EDITORS' PREFACE. 



THE volume called Higher Mathematics, the first edition 
of which was published in 1896, contained eleven chapters by 
eleven authors, each chapter being independent of the others, 
but all supposing the reader to have at least a mathematical 
training equivalent to that given in classical and engineering 
colleges. The publication of that volume is now discontinued 
and the chapters are issued in separate form. In these reissues 
it will generally be found that the monographs are enlarged 
by additional articles or appendices which either amplify the 
former presentation or record recent advances. This plan of 
publication has been arranged in order to meet the demand of 
teachers and the convenience of classes, but it is also thought 
that it may prove advantageous to readers in special lines of 
mathematical literature. 

It is the intention of the publishers and editors to add other 
monographs to the series from time to time, if the call for the 
same seems to warrant it. Among the topics which are under 
consideration are those of elliptic functions, the theory of num- 
bers, the group theory, the calculus of variations, and non- 
Euclidean geometry; possibly also monographs on branches of 
astronomy, mechanics, and mathematical physics may be included. 
It is the hope of the editors that this form of publication may 
tend to promote mathematical study and research over a wider 
field than that which the former volume has occupied. 

December, 1905. 



111 



AUTHOR'S PREFACE. 



This compendium of hyperbolic trigonometry was first published 
as a chapter in Merriman and Woodward's Higher Mathematics. 
There is reason to believe that it supplies a need, being adapted to 
two or three different types of readers. College students who have 
had elementary courses in trigonometry, analytic geometry, and differ- 
ential and integral calculus, and who wish to know something cf the 
hyperbolic trigonometry on account of its important and historic rela- 
tions to each of those branches, will, it is hoped, find these relations 
presented in a simple and comprehensive way in the first half of the 
work. Readers who have some interest in imaginaries are then intro- 
diiced to the more general trigonometry of the complex plane, where 
the circular and hyperbolic functions merge into one class of transcend- 
ents, the singly periodic functions, having either a real or a pure imag- 
inary period. For those who also wish to view the subject in some of 
its practical relations, numerous applications have been selected so as 
to illustrate the various parts of the theory, and to show its use to the 
physicist and engineer, appropriate numerical tables being supplied for 
these purposes. 

With all these things in mind, much thought has been given to the 
mode of approaching the subject, and to the presentation of funda- 
mental notions, and it is hoped that some improvements are discerni- 
ble. For instance, it has been customary to define the hyperbolic 
functions in relation to a sector of the rectangular hyperbola, and to 
take the initial radius of the sector coincident with the principal radius 
of the curve; in the present work, these and similar restrictions are 
discarded in the interest of analogy and generality, with a gain in sym- 
metry and simplicity, and the functions are defined as certain charac- 
teristic ratios belonging to any sector of any hyperbola. Such defini- 
tions, in connection with the fruitful notion of correspondence of points 
on comes, lead to simple and general proofs of the addition-theorems, 
from which easily follow the conversion-formulas, the derivatives, the 
Maclaurin expansions, and the exponential expressions. The proofs 
are so arranged as to apply equally to the circular functions, regarded 
as the characteristic ratios belonging to any elliptic sector. For those, 
however, who may wish to start with the exponential expressions as 
the definitions of the hyperbolic functions, the appropriate order of 
procedure is indicated on page 25, and a direct mode of bringing such 
exponential definitions into geometrical relation with the hyperbolic 
sector is shown in the Appendix. 

December, 1905. 



CONTENTS. 



ART. i. CORRESPONDENCE OF POINTS ON CONICS Page 7 

2. AREAS OF CORRESPONDING TRIANGLES 9 

3. AREAS OF CORRESPONDING SECTORS 9- 

4. CHARACTERISTIC RATIOS OF SECTORIAL MEASURES 10 

5. RATIOS EXPRESSED AS TRIANGLE-MEASURES 10 

6. FUNCTIONAL RELATIONS FOR ELLIPSE n 

7. FUNCTIONAL RELATIONS FOR HYPERBOLA n 

8. RELATIONS BETWEEN HYPERBOLIC FUNCTIONS 12 

9. VARIATIONS OF THE HYPERBOLIC FUNCTIONS 14 

10. ANTI-HYPERBOLIC FUNCTIONS 16 

11. FUNCTIONS OF SUMS AND DIFFERENCES 16 

12. CONVERSION FORMULAS .'..*... 18 

13. LIMITING RATIOS 19 

14. DERIVATIVES OF HYPERBOLIC FUNCTIONS 20 

15. DERIVATIVES OF ANTI-HYPERBOLIC FUNCTIONS 22 

1 6. EXPANSION OF HYPERBOLIC FUNCTIONS 23 

17. EXPONENTIAL EXPRESSIONS 24 

18. EXPANSION OF ANTI-FUNCTIONS 25 

19. LOGARITHMIC EXPRESSION OF ANTI-FUNCTIONS 27 

20. THE GUDERMANIAN FUNCTION 28 

21. CIRCULAR FUNCTIONS OF GUDERMANIAN 28 

22. GUDERMANIAN ANGLE 29 

23. DERIVATIVES OF GUDERMANIAN AND INVERSE 30 

24. SERIES FOR GUDERMANIAN AND ITS INVERSE 31 

25. GRAPHS OF HYPERBOLIC FUNCTIONS 32 

26. ELEMENTARY INTEGRALS ... 35 

27. FUNCTIONS OF COMPLEX NUMBERS . 38 

28. ADDITION THEOREMS FOR COMPLEXES ......... 40 

29. FUNCTIONS OF PURE IMAG IN ARIES . , , . 41 

30. FUNCTIONS OF x+iy IN THE FORM Xi-iY , . . . . . . 43 

31. THE CATENARY . , 47 

32. THE CATENARY OF UNIFORM STRENGTH 49 

33. THE ELASTIC CATENARY 50 

34. THE TRACTORY 51 

35. THE LOXODROME , 52 



6 CONTENTS. 

ART. 36 COMBINED FLEXURE AND TENSION 53 

37. ALTERNATING CURRENTS 55 

38. MISCELLANEOUS APPLICATIONS 60 

39. EXPLANATION OF TABLES 62 

TABLE I. HYPERBOLIC FUNCTIONS 64 

II. VALUES OF COSH (x+iy) AND SINH (x+ iy) 66 

III. VALUES OF gdu AND 6 > 7 

IV. VALUES OF gdw, LOG SINH w, LOG COSH u '7 

APPENDIX. HISTORICAL AND BIBLIOGRAPHICAL 71 

EXPONENTIAL EXPRESSIONS AS DEFINITIONS , 72 

INDEX 73 



HYPERBOLIC FUNCTIONS. 



ART. 1. CORRESPONDENCE OF POINTS ON CONICS. 

To prepare the way for a general treatment of the hyper- 
bolic functions a preliminary discussion is given on the relations 
between hyperbolic sectors. The method adopted is such as 
to apply at the same time to sectors of the ellipse, including 
the circle; and the analogy of the hyperbolic and circular 
functions will be obvious at every step, since the same set of 
equations can be read in connection with either the hyperbola 
or the ellipse.* It is convenient to begin with the theory of 
correspondence of points on two central conies of like species, 
i.e. either both ellipses or both hyperbolas. 

To obtain a definition of corresponding points, let 0^,, 
O l B l be conjugate radii of a central conic, and O^A^ , O^B^ 
conjugate radii of any other central conic of the same species; 
let P lt P^ be two points on the curves; and let their coordi- 
nates referred to the respective pairs of conjugate directions 
be (x^ , jj/,), (x^ , 7,); then, by analytic geometry, 

*2- y L - I *" *" - I (i\ 

i? *,'" ." V" 

* The hyperbolic functions are not so named on account of any analogy 
with what are termed Elliptic Functions. " The elliptic integrals, and thence 
the elliptic functions, derive their name from the early attempts of mathemati- 
cians at the rectification of the ellipse. ... To a certain extent this is a 
disadvantage; . . . because we employ the name hyperbolic function to de- 
note cosh u. sinh u, etc., by analogy with which the elliptic functions would be 
merely the circular functions cos (p, sin </>, etc. . . ."' (Greenhill, Elliptic 
Functions, p. 175.) 



tf HYPERBOLIC FUNCTIONS. 

Now if the points P l , P, be so situated that 

a t a 9 ' b, ~b^ 

the equalities referring to sign as well as magnitude, then P l , 
P t are called corresponding points in the two systems. If Q l , 
Q t be another pair of correspondents, then the sector and tri- 




angle P 1 1 Q 1 are said to correspond respectively with the 
sector and triangle P^O^Q^. These definitions will apply also 
when the conies coincide, the points P l , P a being then referred 
to any two pairs of conjugate diameters of the same conic. 

In discussing the relations between corresponding areas it 
is convenient to adopt the following use of the word " measure": 
The measure of any area connected with a given central conic 
is the ratio which it bears to the constant area of the triangle 
formed by two conjugate diameters of the same conic. 

I r or example, the measure of the sector A^O^P^ is the ratio 

sector A^O.P^ 
triangle A 1 O 1 B 1 



AREAS OF CORRESPONDING SECTORS. 

;and is to be regarded as positive or negative according as 
A 1 O 1 P 1 and A 1 O 1 B 1 are at the same or opposite sides of their 
common initial line. 

ART. 2. AREAS OF CORRESPONDING TRIANGLES. 
The areas of corresponding triangles have equal measures. 
For, let the coordinates of /,, Q l be (>,,/,), (#,',.?,') and ^ et 
those of their correspondents/^, <2 2 be (> 2 , jj/ 2 ), (x*,y)\ let the 
triangles P.O.Q, , P^O^ be T lt T t , and let the measuring tri- 
angles A^O^B^ A^O^B^ be K l , K^, and their angles <, , GO, ; 
then, by analytic geometry, taking account of both magnitude 
and direction of angles, areas, and lines, 

T\ = iCr.j'/ -*,>,) sin QJ = 5 ZL - fi.' A; 
^j #,#! sin GD^ a l , a l , ' 



sn ^a i ^L _ 



A sin 



Therefore, by (2), - = -. (3) 



ART. 3. AREAS OF CORRESPONDING SECTORS. 

The areas of corresponding sectors have equal measures. 
For conceive the sectors S,, S 2 divided up into infinitesimal 
corresponding sectors ; then the respective infinitesimal corre- 
sponding triangles have equal measures (Art. 2) ; but the 
given sectors are the limits of the sums of these infinitesimal 
triangles, hence 



In particular, the sectors A.O.P^ AflJP^ have equal meas- 
ures ; for the initial points A lt A t are corresponding points. 

It may be proved conversely by an obvious reductio ad 
absurdum that if the initial points of two equal-measured 
sectors correspond, then their terminal points correspond. 

Thus if any radii O V A^ O^A^ be the initial lines of two 
equal-measured sectors whose terminal radii are O l P lt Of^ 



10 HYPERBOLIC FUNCTIONS. 

then P lt P 9 are corresponding points referred respectively to 
the pairs of conjugate directions O^A lf O^B^ and 0.,A^ OJ3^\ 
that is, 



Prob. i. Prove that the sector P^O^Q, is bisected by the line 
joining O l to the mid-point of P l Q l . (Refer the points P lt Q iy re- 
spectively, to the median as common axis of x, and to the two 
opposite conjugate directions as axis of y, and show that P 19 Q l 
are then corresponding points.) 

Prob. 2. Prove that the measure of a circular sector is equal to 
the radian measure of its angle. 

Prob. 3. Find the measure of an elliptic quadrant, and of the 
sector included by conjugate radii. 

ART. 4. CHARACTERISTIC RATIOS OF SECTORIAL 

MEASURES. 

Let Aftf^ = S 1 be any sector of a central conic; draw 
P,M^ ordinate to O l A lt i.e. parallel to the tangent at A t ; 
let O l M l = x lt Mf^ = jj>, , O l A l =,, and the conjugate radius 
O l B l = b l ; then the ratios x l /a l , yjb^ are called the charac- 
teristic ratios of the given sectorial measure SJK r These 
ratios are constant both in magnitude and sign for all sectors 
of the same measure and species wherever these may be situ- 
ated (Art. 3). Hence there exists a functional relation be- 
tween the sectorial measure and each of its characteristic 
ratios. 

ART. 5. RATIOS EXPRESSED AS TRIANGLE-MEASURES. 

The triangle of a sector and its complementary triangle are 
measured by the two characteristic ratios. For, let the triangle 
Aflfi and its complementary triangle P 1 O 1 B 1 be denoted by 

T 19 TV; then 

T\ fay i sin 67, , 



l ~ ^a., sn 

7y \b^x^ sin 
K l ~ ^a l b l sin 



FUNCTIONAL RELATIONS FOR ELLIPSE. 



11 



ART. 6. FUNCTIONAL RELATIONS FOR ELLIPSE. 

The functional relations that exist between the sectorial 
measure and each of its characteristic ratios are the same 
for all elliptic, in- Bt 



eluding circular, sec- 
tors (Art. 4). Let/*,, 
P % be corresponding 
points on an ellipse 
and a circle, referred o, 
to the conjugate di- 
rections O.A^ O^BI, and 
right angles ; let the angle 

5, 

K n 





^B^ the latter pair being at 
= in radian measure; then 



'6 



_ 



(6) 



= cos 



= sin 



a 



hence, in the ellipse, by Art. 3, 

l' = cos J-, ^nzsin L. 



(7) 



Prob. 4. Given JCi = \a\; find the measure of the elliptic sector 
.AiOtPi. Also find its area when a^ = 4, , = 3, GJ = 60. 

Prob. 5. Find the characteristic ratios of an elliptic sector whose 
measure is %TT. 

Prob. 6. Write down the relation between an elliptic sector and 
its triangle. (See Art. 5.) 

ART. 7. FUNCTIONAL RELATIONS FOR HYPERBOLA. 

The functional relations between a sectorial measure and 
its characteristic ratios in the case of the hyperbola may be 
written in the form 

x \ u 

-i = cosh 



i .1 , 

'-i = smh _- 



and these express that the ratio of the two lines on the left is 
a certain definite function of the ratio of the two areas on the 
right. These functions are called by analogy the hyperbolic 



12 HYPERBOLIC FUNCTIONS. 

cosine and the hyperbolic sine. Thus, writing u for S t /K lt the 

two equations 

x y 

l . = cosh u, v = sm h u (8V 

a, b, 

serve to define the hyperbolic cosine and sine of a given secto- 
rial measure u ; and the hyperbolic tangent, cotangent, secant, 
and cosecant are then defined as follows : 



smh u coshu 

tanh u = = , coth u = -r 

cosh u sinh u 

r 

sech u = : -, csch u = 



(9) 



cosh 11 sinh u \ 

The names of these functions may be read " h-cosine," 
"h-sine," "h-tangent," etc., or "hyper-cosine," etc. 

ART. 8. RELATIONS AMONG HYPERBOLIC FUNCTIONS. 

Among the six functions there are five independent rela- 
tions, so that when the numerical value of one of the functions 
is given, the values of the other five can be found. Four of 
these relations consist of the four defining equations (9). The 
fifth is derived from the equation of the hyperbola 

V_^_ 

^ b? - 

giving 

cosh* u sinh 2 u = I. (10) 

By a combination of some of these equations other subsidi- 
ary relations may be obtained; thus, dividing (10) successively 
by cosh 2 , sinh 2 &, and applying (9), give 

I tanh 2 u = sech 2 u, ) 

(ii) 
coth 2 u I = csch 2 u. 3 

Equations (9), (10), (11) will readily serve to express the 
value of any function in terms of any other. For example, 
when tanh u is given, 






coth u = - , sech u = \ tanh 2 

tanh u 



RELATIONS BETWEEN HYPERBOLIC FUNCTIONS. 13 

I tanh u 

cosh u = , smh u = , 

V I -- tanh a # V i -- tanh a # 



\ i tanh 2 
cscn u = - 



tanh u t 

The ambiguity in the sign of the square root may usually 
be removed by the following considerations : The functions 
cosh u, sech u are always positive, because the primary char- 
acteristic ratio xja l is positive, since the initial line O l A l and 
the abscissa O l M l are similarly directed from O lt on which- 
ever branch of the hyperbola P l may be situated; but the func- 
tions sinh u, tanh u, coth u, csch u, involve the other charac- 
teristic ratio y l /b l , which is positive or negative according as 
y l and ^ have the same or opposite signs, i.e., as the measure 
u is positive or negative ; hence these four functions are either 
all positive or all negative. Thus when any one of the func- 
tions sinh u, tanh u, csch &, coth u, is given in magnitude and 
sign, there is no ambiguity in the value of any of the six 
hyperbolic functions ; but when either cosh u or sech u is 
given, there is ambiguity as to whether the other four functions 
shall be all positive or all negative. 

The hyperbolic tangent may be expressed as the ratio of 
two lines. For draw the tangent 
line AC=t\ then 

y x ay 

tanh u = <- : - = T . - 

b a b x 

ail' / x i*~ i 1 

= - b .- = - b . ( I2 )0 A M 

The hyperbolic tangent is the measure of the triangle OAC* 
For 

OA C at t 




Thus the sector AOP, and the triangles AOP, FOB, AOC, 
are proportional to z/, sinh #, cosh , tanh u (eqs. 5, 13) ; hence 

sinh u > > tanh u. (14) 



14 



HYPERBOLIC FUNCTIONS. 



Prob. 7. Express all the hyperbolic functions in terms of sinh u. 
Given cosh u = 2, find the values of the other functions. 

Prob. 8. Prove from eqs. 10, u, that coshu> sinh u, coshu>i, 
tanh u < i, sech u < i. 

Prob. 9. In the figure of Art. i, let OA = 2 , OB=i, AOB = 60, 
and area of sector A OP = 3; find the sectorial measure, and the 
two characteristic ratios, in the elliptic sector, and also in the hyper- 
bolic sector; and find the area of the triangle A OP. (Use tables of 
cos, sin, cosh, sinh.) 

Prob. 10. Show that coth u, sech u, csch u may each be ex- 
pressed as the ratio of two lines, as follows: Let the tangent at P 
make on the conjugate axes OA, OB, intercepts OS = m, OT = n\ 
let the tangent at B, to the conjugate hyperbola, meet OP in R y 
making BR = /; then 

coth u = I/ a, sech u = m/a, csch u = n/b. 

Prob. ii. The measure of segment AMP is sinh u cosh u u. 
Modify this for the ellipse. Modify also eqs. 10-14, an d probs. 
8, 10. 

v ART. 9. VARIATIONS OF THE HYPERBOLIC FUNCTIONS. 

^ 

Since the values of the hyperbolic functions depend only 
on the sectorial measure, it is convenient, in tracing their vari- 
ations, to consider only sectors of one 
half of a rectangular hyperbola, whose 
conjugate radii are equal, and to take the 
principal axis OA as the common initial 
line of all the sectors. The sectorial 
measure u assumes every value from oo, 
through o, to -f- oo , as the terminal point 
P comes in from infinity on the lower 
branch, and passes to infinity on the upper 
branch; that is, as the terminal line OP 
swings from the lower asymptotic posi- 
tion y = x, to the upper one, y = x. It is here assumed, 
but is proved in Art. 17, that the sector AOP becomes infinite 
as P passes to infinity. 

Since the functions cosh u, sinh u, tanh u, for any position 




VARIATIONS OF THE HYPERBOLIC FUNCTIONS. 15 

of CP, are equal to the ratios of x, y, /, to the principal radius 
a, it is evident from the figure that 

cosh o = I, sinh = 0, tanh 0=0, (15) 

and that as u increases towards positive infinity, cosh u, sinh u 
are positive and become infinite, but tanh& approaches unity 
as a limit ; thus 

cosh oo = oo , sinh oo = oo , tanh oo = i. (16) 

Again, as u changes from zero towards the negative side, 
cosh u is positive and increases from unity to infinity, but 
sinh u is negative and increases numerically from zero to a 
negative infinite, and tanh u is also negative and increases 
numerically from zero to negative unity ; hence 

cosh ( oo ) = oo , sinh { oo ) = oo , tanh ( oo ) = i. (17) 

For intermediate values of u the numerical values of these 
functions can be found from the formulas of Arts. 16, 17, and 
are tabulated at the end of this chapter. A general idea of 
their manner of variation can be obtained from the curves in 
Art. 25, in which the sectorial measure ?/ is represented by the 
abscissa, and the values of the functions cosh u, sinh u, etc., 
are represented by the ordinate. 

The relations between the functions of // and of u are 
evident from the definitions, as indicated above, and in Art. 8. 
Thus 

cosh ( u) = -f- cosh u, sinh ( 11) = sinh u, \ 

sech (#)=-{- sech , csch ( u) csch u, > (18) 

tanh ( u) = tanh u, coth ( u) = coth u. } 

Prob. 12. Trace the changes in sech u, coth u, csch u, as u passes 
from --oo to + oo. Show that sinh #, cosh u are infinites of the 
same order when u is infinite. (It will appear in Art. 17 that sinh 
u, cosh u are infinites of an order infinitely higher than the order 
of*.) 

Prob. 13. Applying eq. (12) to figure, page 14, prove tanh , = 
tan A OP. 



16 HYPERBOLIC FUNCTIONS. 

ART. 10. ANTI-HYPERBOLIC FUNCTIONS. 

x y t 

The equations -- = cosh u, -= = sinh u, 7 = tanh u, etc., 
a b b 



may also be expressed by the inverse notation u = cosh" 1 



CC- 



y t 

u =: sinh l -7, u = tanh J -T, etc., which may be read: "u is 

the sectorial measure whose hyperbolic cosine is the ratio x to 
a" etc. ; or " u is the anti-h-cosine of x/a" etc. 

Since there are two values of u, with opposite signs, that 
correspond to a given value of cosh u, it follows that if u be 
determined from the equation cosh u = m, where m is a given 
number greater than unity, u is a two-valued function of ;//. 
The symbol cosh' 1 m will be used to denote the positive value 
of u that satisfies the equation cosh u =. m. Similarly the 
symbol sech" 1 ;;/ will stand for the positive value of 21 that 
satisfies the equation sech u m. The signs of the other 
functions sinh' 1 ?;/, tanh' 1 */*, coth" 1 m, csch" 1 ;, are the same 
as the sign of m. Hence all of the anti-hyperbolic functions 
of real numbers are one-valued. 

Prob. 14. Prove the following relations: 

cosh" 1 m = sinh" 1 Vm* - i, sinh~ l m = cosh" 1 riff -j- i, 
the upper or lower sign being used according as /;/ is positive or 
negative. Modify these relations for sin" 1 , cos" 1 . 

Prob. 15. In figure, Art. i,let OA = 2, OB = i,AO = 60; find 
the area of the hyperbolic sector A OP, and of the segment AMP, 
if the abscissa of P is 3. (Find cosh" 1 from the tables for cosh.) 

ART. 11. FUNCTIONS OF SUMS AND DIFFERENCES. 

(a) To prove the difference-formulas 

sinh (u v) = sinh u cosh v cosh u sinh v, ) 

cosh ( v) = cosh u cosh v sinh u sinh v. } 

Let OA be any radius of a hyperbola, and let the sectors A OP, 
AOQ have the measures u, v\ then u -- v is the measure of the 
sector QOP. Let OB, OQ f be the radii conjugate to OA, OQ\ 
and let the coordinates of P, Q, Q be (x l , y^), (x, y), (x', y') 
with reference to the axes OA, OB\ then 



FUNCTIONS OF SUMS AND DIFFERENCES. 



17 



- U 
smh 



_ ) = sinh ?^LQOP = triangle QOP ^ ^ 
K. AT 



sn 



a)^ sn &? ^ a l l a t 

= sinh u cosh z; cosh u sinh z/ ; 



k <2' 





cosh (H z>) = cosh 



sector QOP triangle POQ' 
- 



K. 



K 



. 

[Art. 5, 



sn 



sn 



but 



(20) 



since Q, Q' are extremities of conjugate radii ; hence 
cosh (u v) cosh u cosh v sinh u sinh v. 

In the figures u is positive and v is positive or negative. 
Other figures may be drawn with u negative, and the language 
in the text will apply to all. In the case of elliptic sectors, 
similar figures may be drawn, and the same language will apply, 
except that tlie second equation of (20) will be x r /a l = 
therefore 

sin (u v) = sin u cos v cos u sin v t 

cos (u v) = cos u cos v -f- sin u sin v. 

(b) To prove the sum-formulas 

sinh (u -f- v) = sinh u cosh v -f- cosh u sinh v, ) 
cosh (u -f- z>) cosh ?/ cosh z> -f- sinh u sinh z;. ) 

These equations follow from (19) by changing v into zv 



(21) 



18 HYPERBOLIC FUNCTIONS. 

and then for sinh ( v), cosh ( v) y writing sinh v, cosh v 
(Art. 9, eqs. (18)). 

(c) To prove that tanh (uv\ = tgg*Ltanhp ( 22 ) 

i tanh tanh? 

Writing tanh (u v) = sin ( u v ) expanding and dividing 

cosh (ii _ v) 

numerator and denominator by cosh u cosh z>, eq. (22) is ob- 
tained. 

Prob. 1 6. Given cosh u 2, cosh v 3, find cosh (u -f- z>). 
Prob. 17. Prove the following identities: 

1. sinh 2U 2 sinh # cosh w. 

2. cosh 2u cosh 2 ?/ -|- sinh a i -}- 2 sinh 2 u = 2 cosh 3 u \. 

3. i -h cosh u 2 cosh 2 #, cosh & -- i = 2 sinh 2 -JT&. 

sinh z; cosh u i /cosh i\* 

4. tanh \u - - = - - = 

i -j- cosh u smh u \cosh u -f- i/ 

. 2 tanh & T -L- tanh 2 & 

5. smh 2U = - cosh aw 



. 
i tanh u i - tanh & 

6. sinh 3^ 3 sinh u -j- 4 sinh 3 #, cosh 3// = 4 cosh 3 // 3 cosh w. 

i -f tanh \u 

7. cosh u 4- smh w 

i tanh \u 

8. (cosh u -\- sinh ?/)(cosh z; + sinh ^) = cosh (// -f v) -\- sinh (?/ -f- T). 

9. Generalize (8); and show also what it becomes when u=v= . . , 
10. sinh 2 .*: cos'jy -f- cosh 2 .* sinV sinhV + sin 2 /. 

IT. cosh" 1 *?/ cosh~ l n = cosh" 1 ^;;/;/ y (w 2 - i)( a 



12. sinh l m sinh l n = sinh *\m \ T. -\- n* ny i -f- 

Prob. 18. What modifications of signs are required in (21), (22), 
in order to pass to circular functions ? 

Prob. 19. Modify the identities of Prob. 17 for the same purpose. 

ART. 12. CONVERSION FORMULAS. 

To prove that 

cosh u^-\- cosh z/ a = 2 cosh J(^+ & 2 ) cosh \(u^ ?/ 2 ), 

cosh , cosh 2/ 2 = 2 sinh (#, -(- ?/ Q ) sinh %(u u\ 

(21} 
sinh ?/, -f- sinh ?/. 2 = 2 sinh (#, -(- ;/,) cosh \ r u^ u 9 \ 

sinh u, sinh a = 2 cosh i(?/. 4- ?/) sinh ^(u ). 

- \ i a v i "^ J 



LIMITING RATIOS. 19 

From the addition formulas it follows that 

cosh (u -\- v) -f- cosh (u v) = 2 cosh u cosh v y 
cosh (u -\- v) cosh (u v) = 2 sinh u sinh v, 
sinh ( + ^) + sinh ( v) = 2 sinh cosh z>, 
sinh (w + v) -- sinh (u -- v) = 2 cosh u sinh v, 

and then by writing u + v = u l , u v u^, u =; J(a, + #,), 
z/ = (?/, a ), these equations take the form required. 

Prob. 20. In passing to circular functions, show that the only 
modification to be made in the conversion formulas is in the alge- 
braic sign of the right-hand member of the second formula. 

cosh 2 + cosh AV cosh 2U -f- cosh 4^ 

Prob. 21. Simplify -^-. -- ; . -- , - r r - 

sinh 2U -f- sinh 4z; cosh 2U cosh $v 

Prob. 22. Prove sinh 9 ,* sinh 2 ^ = sinh (x ~{-y) sinh (x y)* 
Prob. 23. Simplify cosh 2 .* cosh'jy sinh 2 A: sinh 2 ^. 
Prob. 24. Simplify cosh 2 jc cos 2 )' -f- sinh 2 ^ si 



ART. 13. LIMITING RATIOS. 
To find the limit, as u approaches zero, of 

sinh u tanh u 

> > 

u u 

which are then indeterminate in form. 

By eq. (14), sinh u > u > tanh u ; and if sinh u and tanh u 
be successively divided by each term of these inequalities, it 
follows that 

sinh u 

I < - -- < cosh u, 
u 

tanh u 

sech u < - < i; 
u 

but when u-^o, cosh w = I, sech u i, hence 

lim. si" h K __ i im . tanh a _ ^ , . 

u = o u u db o ^ 



HYPERBOLIC FUNCTIONS. 



ART. 14. DERIVATIVES OF HYPERBOLIC FUNCTIONS. 
To prove that 

/ x d[sinh u} 

(*) - 1 = cosh u, 

du 



(*) 

<*) 

(</) 



d(cosh 
du 



du 



u) 



du 
/ x d(coth w) 



(/) 



du 
d[csch ) 
du 



= sinh #, 
= sech s # t 

sech u tanh w, 

csch 2 w, 

= csch u coth u. 



(a) Let j = sinh u, 

Ay = sinh ( -f~ ^) ~~ smn u 
= 2 cosh (2& -f~ Au} sinh 
sinh 



-p- = cosh (u -f 
Au 

Take the limit of both sides, as Au ~ o, and put 

Ay dy dVsinh u) 
lim. - - = ~r = ~ j --- 



lim. 

then 

Similar to (<7). 



lim. cosh ( -f~ JzJ) = cosh ?/, 
sinh 



(see Art. 13) 



du 



sinh u 



= cosh w. 



du ' cosh # 
cosh 2 u sinh 9 
cosh 8 u 



(25) 



cosh* // 



= sech*. 



DERIVATIVES OF HYPERBOLIC FUNCTIONS. 21 

(d) Similar to (c). 

*/(sech u] d I sinh u 

(e) . = -7- \ = n~ = seen u tanh u. 

du au cosh u cosh u 

(f) Similar to (e). 

It thus appears that the functions sinh u, cosh u reproduce 
themselves in two differentiations ; and, similarly, that the 
circular functions sin u, cos u produce their opposites in two 
differentiations. In this connection it may be noted that the 
frequent appearance of the hyperbolic (and circular) functions 
in the solution of physical problems is chiefly due to the fact 
that they answer the question : What function has its second 
derivative equal to a positive (or negative) constant multiple 
of the function itself? (See Probs. 28-30.) An answer such as 
y = cosh mx is not, however, to be understood as asserting that 
mx is an actual sectorial measure and y its characteristic ratio ; 
but only that the relation between the numbers mx and y is the 
same as the known relation between the measure of a hyper- 
bolic sector and its characteristic ratio ; and that the numerical 
value of y could be found from a table of hyperbolic cosines. 

Prob. 25. Show that for circular functions the only modifica- 
tions reqwired are in the algebraic signs of (b), (d). 

Prob. 26. Show from their derivatives which of the hyperbolic 
and circular functions diminish as u increases. 

Prob. 27. Find the derivative of tanh u independently of the 
derivatives of sinh , cosh u. 

Prob. 28. Eliminate the constants by differentiation from the 
equation j/ = A cosh mx -\- B sinh mx, and prove that d*y'/d& = m*y. 

Prob. 29. Eliminate the constants from the equation 
y = A cos mx -f- B sin mx, 

and prove that d*y/dx* = my. 

Prob. 30. Write down the most general solutions of the differen- 
tial equations 



22 HYPERBOLIC FUNCTIONS. 

ART. 15. DERIVATIVES OF ANTI-HYPERBOLIC FUNCTIONS. 



(*) 
w 
w 
w 

z; = 


^(sinh- 1 


*) ' 


(26) 

= cosh ^/ 


dx 


V^f 2 -4- i 7 
^r) _ .1 


dx 
</(tanh~ ] 


x}_ i -J 


dx 
^(coth- 1 


*) I 1 


dx 


*)_ I 


dx 
^/(csch- 1 


* Vl - x* 
*}- I 


dx 
sinh" 1 ;r, 


x Vx* + i 
then x = sinh w, ^r 



= Vi + sinh 2 </ = iT+1? </#, ^ = ^/ Vi -j- ^ a . 
(3) Similar to (a). 

(c) Let # = tanh" 1 x, then # = tanh u, dx sech 3 u du 
= (i tanh 2 #)d = (i x*)du, du = dx/i x\ 

(d) Similar to (c). 

dx ~ dx\ x'~ x* I W a / x Vi x* 

(/) Similar to (e). 

Prob. 31. Prove 



4- 



EXPANSION OF HYPERBOLIC FUNCTIONS. 23 

Prob. 32. Prove 

,.,_,* dx j <u-i x dx 

a sinh =- , a; cosh = == 

a Vx* -\- a* a Vx* a" 

. x adx ~~| . x adx 

^tanh- - = -. E , ^coth" 1 -= * 

a a x _\x<a a x - a J*> 

Prob. 33. Find ^(sech" 1 x) independently of cosh" 1 x. 

Prob. 34. When tanh" 1 x is real, prove that coth" 1 x is imagi- 
nary, and conversely; except when x = i. 

sinh~ l x cosh" 1 x 

Prob. 35. Evaluate , , when ,# = 00. 

log x log x 

ART. 16. EXPANSION OF HYPERBOLIC FUNCTIONS. 

For this purpose take Maclaurin's Theorem, 

/(a) = /(o) + uf(o) + y"(o) + p. /'"(o) + . . ., 

z - 6- 

and put f(u) = sinh u, f(u] = cosh u, f"(u) = sinh #,..., 

then /(o) = sinh = 0, /'(o) = cosh o = I, . . .; 

hence sinh u = u -\-u* -\ 1 u 6 + . . . ; (27) 

and similarly, or by differentiation, 

cosh u = i-l r^'H r^ 4 + (28) 

2! 4! 

By means of these series the numerical values of sinh u, 
cosh u, can be computed and tabulated for successive values of 
the independent variable u. They are convergent for all values 
of u, because the ratio of the th term to the preceding is in 
the first case u*/(2n i)(2n 2), and in the second case 
u*/(2n 2}(2n 3), both of which ratios can be made less than 
unity by taking n large enough, no matter what value u has. 
Lagrange's remainder shows equivalence of function and series^ 



24 HYPERBOLIC FUNCTIONS. 

From these series the following can be obtained by division : 

tanh u = u %u s -f- T 2 T w 5 4- ^feu" 1 -\- . . . , 
sech u = I - J" + ^ - ^u 6 + . . . , 



u coth = 
it csch a = i 



(29) 



These four developments are seldom used, as there is no 
observable law in the coefficients, and as the functions tanh , 
sech u, coth u, csch u, can be found directly from the previously 
computed values of cosh u, sinh u. 

Prob. 36. Show that these six developments can be adapted to 
the circular functions by changing the alternate signs. 

ART. 17. EXPONENTIAL EXPRESSIONS. 
Adding and subtracting (27), (28) give the identities 

cosh u 4- sinh u = i -\- u -\- u* -I -- 7 u* -4- -u* 4- . . = e u , 

2\ 3! 4! 

cosh u sinh u I u -\ -- -u* -- -u 3 4- -u* . . = e~ u . 

2\ 3! 4! 

hence cosh u = \(e u -f- e~ u ), sinh u = %(e u e~ u ), 1 

e * _ e -n 2 r (3) 

tanh u - -- , sech u = --- , etc. i 
P + e~ H e" + e- u } 

The analogous exponential expressions for sin u, cos u are 
cos u = \e ui -\-e~ ui \ sin u = (e ui e~ ni \ (i = V i) 

22 

where the symbol e ui stands for the result of substituting ui for 
x in the exponential development 



This will be more fully explained in treating of complex 
numbers, Arts. 28, 29. 



EXPANSION OF ANTI-FUNCTIONS. #5 

Prob. 37. Show that the properties of the hyperbolic functions 
could be placed on a purely algebraic basis, by starting with equa- 
tions (30) as their definitions ; for example, verify the identities : 

sinh ( u) = sinh u, cosh ( u) = cosh u, 
cosh 2 u sinh 2 u i , sinh (u -f- v) = sinh u cosh v ~f- cosh u sinh v 9 

</ 2 (cosh mu) </*(sinh mu) 

L m cosh /#, - = nr sinh w#. 

du du 

Prob. 38. Prove (cosh u -f- sinh &) M = cosh mi -f- sinh nu. 

Prob. 39. Assuming from Art. 14 that cosh u, sinh u satisfy the 
differential equation d*y/du* = y, whose general solution may be 
written y = Ae u -j- Be~ u , where A, B are arbitrary constants ; show 
how to determine A,jB'm order to derive the expressions. for cosh u y 
sinh u, respectively. [Use eq. (15).] 

Prob. 40. Show how to construct a table of exponential func- 
tions from a table of hyperbolic sines and cosines, and vice versa. 

Prob. 41. Prove u = log,, (cosh u -\- sinh u}. 

Prob. 42. Show that the area of any hyperbolic sector is infinite 
when its terminal line is one of the asymptotes. 

Prob. 43. From the relation 2 cosh u e u -f- e~ H prove 



2 M ~ 1 (cosh #) M =cosh nu + ncosh (n2)u+^n(ni) cosh (n 4)0 + . . ., 

and examine the last term when n is odd or even. 

Find also the corresponding expression for 2 n ~ l (sinh u) n . 



ART. 18. EXPANSION OF ANTI-FUNCTIONS. 

. dfsinrr 1 x) i 

Since -2 -= -- '- == = (I 4- 
dx 



- ----.. 

2 24 246 

hence, by integration, 

i x* i 3 x" i 3 5 x" 1 

sinh- 1 * = * - -+..., (31) 

23 245 2467 

the integration-constant being zero, since sinh' 1 x vanishes 
with x. This series is convergent, and can be used in compu- 



26 HYPERBOLIC FUNCTIONS. 

tation, only when x < I. Another series, convergent when 
x > i, is obtained by writing the above derivative in the form 



= (^ + ,)-* = , - 

dx } 



-if"] -II + III. 135 I 1 

" A. 2X* 2S 26^ "' 



.-. sinh- 1 * = C+log *+-' l^-ll J +13 | ' (32) , 

2 2X* 2 4 4X* 2 466** 

where C is the integration-constant, which will be shown in 
Art. 19 to be equal to log e 2. 

A development of similar form is obtained for cosh" 1 x\ for 



1 x) I/ l\ 

2 -- - 



~* 



dx x 

+ ii + 13 J. , I 3 5 , 

2^ 2 2 4 ^ 2 4 6; 6 ^ '' 

hence 



cosh- 1 x= , 

2 2^ 2 2 44*' 246 6x* 

in which 7 is again equal to log,, 2 [Art. 19, Prob. 46]. Im 
order that the function cosh' 1 ^ maybe real, x must not be- 
less than unity ; but when x exceeds unity, this series is con- 
vergent, hence it is always available for computation. 



x 
Again, 



..., 

and hence tanh' 1 ^^^-) x*-\--x*-\ x 1 + ..., (34) 

j / 

From (32), (33), (34) are derived : 
sech" 1 x = cosh" 1 



x 



2.2 2.4.4 2.4.6.6 



LOGARITHMIC EXPRESSION OF ANTI-FUNCTIONS. 27 

i I I II ! 3 ! !35I 

csch-' x = smh- 1 - = +-- - + ..., 

X X 2 $X 2 4 $X 246 Jx" 

~ 1 x* i . 3 . x* i . 3 . 5 . x* 

= C \ogx-\-- - ...; (36) 

2.2 2.4.4 ' 2.4.6.6 

coth-' x = tanh- 1 I = I + -i- -f -L -L _L + . . .. (37) 

x x 3*' $x b ?S 

Prob. 44. Show that the series for tanh" 1 x, coth" 1 x, sech" 1 x, 
are always available for computation. 

Prob. 45. Show that one or other of the two developments of the 
inverse hyperbolic cosecant is available. 

ART. 19. LOGARITHMIC EXPRESSION OF ANTI-FUNCTIONS. 



Let x = cosh u, then Vx* - i = sinh u\ 



therefore x 4- Vx* - i == cosh u -j- sinh u = e u , 

and u, = cosh ~ l x, = log (x -f- l/jr 2 - i). (38) 

Similarly, sintrttr = log (# -(- i/^r* -f- i). (39) 



Also sech" 1 ^ = cosh" 1 - = log It (40) 

x x 



i i _J_ i/i _i_ ^ a 

cscrrttr = sinh" 1 - = log - . (41) 

^M >- 

Again, let x = tanh u = , 



therefore - = ^- = <? 2 *, 
i ;tr e * 

I 4- # i 4-^r 

^ ^y - I ^\f"f 1~ Q f> Kl ^* JL, I ^\ Qp / J SJ \ 

3 i -- x ' I x 

I ^f I I 

and coth" 1 :*; = tanh' 1 - = J log - -. (43) 

^r jir i 

Prob. 46. Show from (38), (39), that, when x = oo, 

sinh"" 1 ^: log ^ ^ log 2. cosh" ^ ^ - log jc -i log 2, 

and hence show that the integration-constants in (32), (33) are each 
<equal to log 2. 



28 HYPERBOLIC FUNCTIONS. 

Prob. 47. Derive from (42) the series for tanh' 1 .* given in (34). 
Prob. 48. Prove the identities: 

x i oc* i 

log.x=2tanh~ 1 - =tanh -1 -=sinh~ J (*" ^~ 1 )= 
x+i x +i 



log sec x = 2 tanh' 1 tan 2 \x; log esc x = 2 tanh" ! tan a (i^r -f- $x)\ 
log tan x = tanh' 1 cos 2^" sinh" 1 cot 2x = cosh" 1 esc 2x. 

ART. 20. THE GUDERMANIAN FUNCTION. 

The correspondence of sectors of the same species was dis- 
cussed in Arts. 1-4. It is now convenient to treat of the 
correspondence that may exist between sectors of different 
species. 

Two points PV P^ , on any hyperbola and ellipse, are said to 
correspond with reference to two pairs of conjugates O t A l9 
O^B t , and O n A^ , OJB* , respectively, when 

xja, = a,/x v (44). 

and when y lt y t have the same sign. The sectors A^O.P^ 
A^O^P^ are then also said to correspond. Thus corresponding 
sectors of central conies of different species are of the same 
sign and have their primary characteristic ratios reciprocal. 
Hence there is a fixed functional relation between their re- 
spective measures. The elliptic sectorial measure is called 
the gudermanian of the corresponding hyperbolic sectorial 
measure, and the latter the anti-gudermanian of the former. 
This relation is expressed by 

S,/K, = gd S,/K, 

or v = gd u, and u = gd" 1 ^. (45) 

ART. 21. CIRCULAR FUNCTIONS OF GUDERMANIAN. 

The six hyperbolic functions of u are expressible in terms 
of the six circular functions of its gudermanian ; for since 

^-= cosh , = cos v, (see Arts. 6, 7) 

a, a, 

in which u, v are the measures of corresponding hyperbolic 
and elliptic sectors, 



GUDERMANIAN ANGLE. 29 1 

hence cosh u = sec v, [eq. (44)] 



sinh u = V sec 2 ^ I = tan v, 
tanh u = tan ^/sec v = sin z/, 
coth w = esc z/, 
sech u = cosz/, 
csch & = cot v. 

The gudermanian is sometimes useful in computation ; for 
instance, if sinh u be given, v can be found from a table of 
natural tangents, and the other circular functions of v will give 
the remaining hyperbolic functions of u. Other uses of this 
function are given in Arts. 22-26, 32-36. 

Prob. 49. Prove that gd u = Bec l ~ 1 (cosh u) = tan~ 1 (sinh u) 

= cos'^sech u) =sin~ 1 (tanh u), 

Prob. 50. Prove gd" 1 ^ = cosh'^sec v) = sinh" 1 (tan v) 

= sech" 1 (cos v) = tahh -1 (sin v). 

Prob. 51. Prove gd o = o, gd oo = ^TT, gd( oo) = ^7f y 

Prob 52. Show that gd u and gd" 1 v are odd functions of u, v. 

Prob. 53. From the first identity in 4, Prob. 17, derive the rela- 
tion tanh \u = tan \v. 

Prob. 54. Prove 

tanh" J (tan )= gd 2u y and tan -1 (tanh x) = -J gd~ ! 2^. 

ART. 22. GUDERMANIAN ANGLE 

If a circle be used instead of the ellipse of Art. 20, the 
gudermanian of the hyperbolic sectorial measure will be equal 
to the radian measure of the angle of the corresponding circular 
sector (see eq. (6), and Art. 3, Prob. 2). This angle will be 
called the gudermanian angle ; but the gudermanian function v y 
as above defined, is merely a number, or ratio ; and this number 
is equal to the radian measure of the gudermanian angle 0, 
which is itself usually tabulated in degree measure ; thus 

6 = iSov/7r (47) 



:30 



HYPERBOLIC FUNCTIONS. 




Prob. 55. Show that the gudermanian angle of u may be construct- 
ed as follows: 

Take the principal radius OA of an equilateral hyperbola, as the 

initial line, and OP as the terminal 
line, of the sector whose measure is #; 
from M, the foot of the ordinate of 
P, draw MT tangent to the circle 
whose diameter is the transverse axis; 
then AO T is the angle required.* 

Prob. 56. Show that the angle 9 
never exceeds 90. 

Prob. 57. The bisector of angle AO T 
M bisects the sector A OP (see Prob. 13, 
Art. 9, and Prob. 53, Art. 21), and the line AP. (See Prob. i, Art. 3.) 
Prob. 58. This bisector is parallel to TP, and the points T, P 
are in line with the point diametrically opposite to A. 

Prob. 59. The tangent at P passes through the foot of the 
.ordinate of T, and intersects TM on the tangent at A. 

Prob. 60. The angle APM is half the gudermanian angle. 

ART. 23. DERIVATIVES OF GUDERMANIAN AND INVERSE. 

Let v = gd u, u = gd" 1 v, 

then sec v = cosh u, 

sec v tan vdv = sinh u du, 

sec vdv = du, 
therefore ^(gd- 1 v) sec vdv. (48) 

Again, dv = cos v du = sech u du, 

therefore ^(gd u) = sech u du. (49) 

Prob. 61. Differentiate: 

y = sinh u gd u, y = sin v -f- gd" 1 v, 

y = tanh u sech u -j- gd u, y = tan v sec v -\- gd" 1 v. 

* This angle was called by Gudermann the longitude of u, and denoted by lu. 
His inverse symbol was ft; thus u = H(/). (Crelle's Journal, vol. 6, 1830.) 
Lambert, who introduced the angle 6, named it the transcendent angle. (Hist, 
de 1'acad, roy de Berlin, 1761). Hotiel (Nouvelles Annales, vol. 3, 1864) 
called it the hyperbolic amplitude of u, and wrote it amh u, in analogy with the 
amplitude of an elliptic function, as shown in Prob. 62. Cayley (Elliptic 
Functions, 1876) made the usage uniform by attaching to the angle the name 
of the mathematician who had used it extensively in tabulation and in the 
,heory of elliptic functions of modulus unity 



SERIES FOR GUDERMANIAN AND ITS INVERSE. 31 

Prob. 62. Writing the "elliptic integral of the first kind" in 
the form p d(f) 

u = 




Vi K* sin 8 0' 

K being called the modulus, and the amplitude; that is, 

= am u, (mod. /c), 
show that, in the special case when K = i, 

u = gd~ l 0, am u = gd u, sin am u = tanh #, 

cos am u = sech u, tan am u = sinh u\ 

and that thus the elliptic functions sin am u, etc., degenerate into 
the hyperbolic functions, when the modulus is unity.* 

ART. 24. SERIES FOR GUDERMANIAN AND ITS INVERSE. 

Substitute for sech u, sec v in (49), (48) their expansions, 
Art. 16, and integrate, then 

gd u = u - K + jV - jfofU 1 + . . . (50) 

gd-'z, = v + X + Jr*' +TO</ + . . . (51) 

No constants of integration appear, since gd u vanishes with 
u, and gd~V with ^. These series are seldom used in compu- 
tation, as gd u is best found and tabulated by means of tables 
of natural tangents and hyperbolic sines, from the equation 

gd u = tan -1 (sinh u), 

and a table of the direct function can be used to furnish the 
numerical values of the inverse function ; or the latter can be 
obtained from the equation, 

gd" 1 ^ = sinh~'(tan v) = cosh'^sec ^). 
To obtain a logarithmic expression for gd~X let 
gd" 1 ^ = u, v gd u, 

* The relation gd u = am u, (mod. i), led Hoiiel to name the function gd u t 
the hyperbolic amplitude of u, and to write itamh u (see note, Art. 22). In this 
connection Cayley expressed the functions tanh u, sech u, sinh u in the form 
sin gd u, cos gd u, tan gd -u, and wrote them sg u, eg u, tg u, to correspond 
with the abbreviations sn u, en u, dn u for sin am u, cos am u, tan am u. 
Thus tanh u = sg u = sn u, (mod. i); etc. 

It is well to note that neither the elliptic nor the hyperbol'c functions 
received their names on account of the relation existing between them in a 
special case. (See foot-note, p. 7 ) 



32 
therefore 



HYPERBOLIC FUNCTIONS. 



sec v = cosli u, tan v = sinh 
sec v -j- tan v = cosh u -\- sinh & = e", 
I -\- sin v I cos (^TT 

f^ \ & 

cos v sin (TT -j- 

&, = gd l v, = log, tan 
gd u u 



= tan 



Prob. 63. Evaluate 



u 



gd v v 



_]v=o 



(52) 



Prob. 64. Prove that gd u sin u is an infinitesimal of the fifth 
order, when u = o. 

Prob. 65. Prove the relations 
j7T + ^= tan' 1 **, i?r iz> 



^/ ART. 25. GRAPHS OF HYPERBOLIC FUNCTIONS. 

Drawing two rectangular axes, and laying down a series of 
points whose abscissas represent, on any convenient scale, suc- 
cessive values of the sectorial measure, and whose ordinates 
represent, preferably on 
the same scale, the corre- 
sponding values of the 
function to be plotted, the 
locus traced out by this 
series of points will be a 
graphical representation of 
the variation of the func- 
tion as the sectorial meas- 





GRAPHS OF THE HYPERBOLIC FUNCTIONS. 33 

ure varies. The equations of the curves in the ordinary carte- 
sian notation are : 

Fig. Full Lines. Dotted Lines. 

A y = cosh x, y = sech x ; 

B y = sinh x, y csch x ; 
C y = tanh x, y coth x ; 
D y = gd x. 

Here x is written for the sectorial measure u, and y for the 
numerical value of cosh u, etc. It is thus to be noted that the 
variables x, y are numbers, or ratios, and that the equation 
y cosh x merely expresses that the relation between the 
numbers x and y is taken to be the same as the relation be- 
tween a sectorial measure and its characteristic ratio. The 
numerical values of cosh u, sinh u, tanh u are given in the 
tables at the end of this chapter for values of u between o and 
4. For greater values they may be computed from the devel- 
opments of Art. 16. 

The curves exhibit graphically the relations : 

i 

sech u - , csch u - , coth u = ; 

cosh u sinh u tanh u 

cosh u < I, sech u > I, tanh u > I, gd u < J?r, etc. ; 
sinh ( ) = sinh ?/, cosh ( u) = cosh u, 
tanh ( u) = tanh u, gd ( u} = gd u, etc.; 
cosh o = I, sinh o o, tanh = 0, csch (o/ =00 , etc.; 
cosh ( oo ) = oo, sinh ( oo ) = 00, tanh ( 00)= i,etc. 

The slope of the curve y = sinh x is given by the equation 
dy/dx = cosh #, showing that it is always positive, and that 
the curve becomes more nearly vertical as x becomes infinite. 
Its direction of curvature is obtained from d^y/dx* sinh x, 
proving that the curve is concave downward when x is nega- 
tive, and upward when x is positive. The point of inflexion is 
at the origin, and the inflexional tangent bisects the angle 
between the axes. 



34 



HYPERBOLIC FUNCTIONS. 



The direction of curvature of the locus y = sech x is given 
by d*y/dx* = sech x(2 tanh 2 x i), and thus the curve is con- 
cave downwards or upwards 

\ according as 2 tanh 2 x i is 

\ negative or positive. The in- 

*** flexions occur at the points 

x tanh- 1 .707, = .881, 
y = .707 ; and the slopes of 
j the inflexional tangents are 



-i-- 




that x is so small as .1. 



The curve y = csch x is 
asymptotic to both axes, but 
approaches the axis of x more 
rapidly than it approaches the 
axis of y, for when x = 3, y is 
only .1, but it is not till/ 10 
The curves y = csch x, y = sinh x 



cross at the points x = .881, y = i. 




Prob. 66. Find the direction of curvature, the inflexional tan- 
gent, and the asymptotes of the curves y = gda% y = tanh x. 

Prob. 67. Show that there is no inflexion-point on the curves 
y = cosh x, y = coth x. 

Prob. 68. Show that any line y = mx -j- n meets the curve 
y = tanh x in either three real points or one. Hence prove that 
the equation tanh x = mx -\- n has either three real roots or one. 
From the figure give an approximate solution of the equation 
tanh x = x i. 



ELEMENTARY INTEGRALS. 35 

Prob. 69. Solve the equations: cosh x x -f- 2; sinh x = |^; 
gd ^ = x J-TT. 

Prob. 70. Show which of the graphs represent even functions, 
and which of them represent odd ones. 



ART. 26. ELEMENTARY INTEGRALS. 

The following useful indefinite integrals follow from Arts. 
14, 15. 23: 

Hyperbolic. Circular. 

1. / sinh u du = cosh u, I sin u du = cos u, 

2. I cosh u du = sinh u, i cos u du = sin u, 

3. /tanh ;; dfo = log cosh ti, I tan udu =^ log cos , 

4. / coth u du = log sinh u, j cot & d# = log sin u, 

5. f csohudu = log tanh , / esc u du = log tan , 
J 2 v 2 

= sinh-^csch u), = -- cosh-'(csc u), 

6. / sech u du = gd u, I sec u du =. gd- 1 u, 

C dx x r dx x 

7' J ,/-T-7 i = S1Ilh ~ - * J ~^= =* = S1 ' n ~ -> 

x a 



f dx x r -dx x 

J 7^ ^ = C a ' J ~^T ^ = C S " a' 

r *& ~~~ M* r c/' " "~ -^ c*- 

x T dx I . ^r 



/ ^r i , ,x r 

9. / - 9 a = -tanh- -, / - 

v d % | ft ft fc u, 



-\-x* t 

I " / ^ 

* Forms 7-12 are preferable to the respective logarithmic expressions 
(Art. 19), on account of the close analogy with the circular forms, and also 
because they involve functions that are directly tabulated. This advantage 
appears more clearly in 13-20. 



36 HYPERBOLIC FUNCTIONS. 



/__^J- J /* fix T , 

-1 = -COth- 1 -, / zrr-COt- 1 -, 

^ a^ x>a a a *s a* -\- x* a a 



dx i x r dx i x 

~ '' a 



/ dx i .x 

. =-csch~ , 

-*- A//I* _J_ -*- a /Z /7 



dx i . x [* - dx i ;r 

12. / , , . , =-csch- . / .,-- =- csc- 1 - 



a x < a a 



From these fundamental integrals the following may be 
derived : 

r dx i ax-\-b 

13. / x , = 7=sinh~" -. - , a positive. ac~>b \ 

Vox* + 2fa + c Va Vac-P 



i . ax -4- b 

= =cosh~ _ , ^positive, ac 

y^ VPac 

l ax^-b 

= , - cos- , a negative. 



/dx 
- 



I ^^r 4- b 

- tanh " 



ac \ o ac 

i ax -\- b 

coth ~ = =f - ac 



ac ac 



Thus, /%r-^- ;=-^coth- l (^-2)l= 
/4 -^ "4 ar ~r3 _U 



coth- 1 2-coth- 1 3 



/* ^ n 2 - 5 

-,- ^T- =-tanh- 1 (^-2) =tanh- 1 o-tanh- 1 (.5) 
M ^r 4^r--j _J 2 



- -5494- 

(By interpreting these two integrals as areas, show graph- 
ically that the first is positive, and the second negative.) 



^ 2 /x b 

- tanh~ 



*For tanh -1 (.5) interpolate between tanh (.54) = 4930, tanh (.56) = . 50*0 
(see tables, pp. 64., 65); and similarly for t.mh- 1 (.",333). 



ELEMENTARY INTEGRALS. 37 



Xb 2 X b 

\ / ~L i or / COth" A / - T- ; 

Y b a Vab V a ~ b 

the real form to be taken. (Put x b = z*, and apply 9, 10.) 



2 

or . tan 

Vb a 



p dx 2 bx 

16. I - = tanh- 1 \ / -, , 

J (ax)Vbx Vba \ b-a 

2 / b X 2 

or 



Vba 



bx 2 bx 

coth- 1 A / , or tan - \ / - ; ; 

V b a' Va b \ <*& 



the real form to be taken. 

/i I i I X 

2 2 a 

By means of a reduction-formula this integral is easily made 
to depend on 8. It may also be obtained by transforming 
the expression into hyperbolic functions by the assumption 
x = a cosh u, when the integral takes the form 

/a* / I 

sinh a udu= / (cosh 2u i)du = - 2 (sinh 2u 2?/) 

= Jtf 2 (sinh u cosh u u), 

which gives 17 on replacing a cosh u by x, and # sinh u by 
(;r 2 # 2 )i The geometrical interpretation of the result is 
evident, as it expresses that the area of a rectangular-hyper- 
bolic segment AMP is the difference between a triangle OMP 
and a sector OAP. 

1 8. f(a* - x*?dx = -x(c? - # f )* + -a* sin" 1 -. 
J x 2 v 2 a 

/\ I i I .2T 

f^ a I cf^dx ^"fjtr 2 [ # 2 ^ 5 [ ^2* sinh -1 . 
' 2 2 a 



20. Aec 3 <pd(j) =f(i + tan 2 0)V tan 

= i tan 0(i + tan 2 0) 1 -f ^ sinh" 1 (tan 0) 
= i sec tan + J gd' 1 0. 

21. / sech 3 w^w= ^ sech u tanh ^ + i gd ?/. 

Prob. 71. What is the geometrical interpretation of 18, 19? 
Prob. 72. Show thaty (tfjc 2 -{- 2^0: + ^)^ reduces to 17. 18, 19, 



38 H\PERBOLIC FUNCTIONS. 

respectively: when a is positive, with ac < b* ; when a is negative ; 
and when a is positive, with ac > ^ 2 . 

Prob. 73. Prove / sinh u tanh u du =. sinh # gd w, 

/cosh & coth u du cosh & -f- log tanh . 
2 

Prob. 74. Integrate 

Prob. 75. In the parabola y 2 = 4px, if $" be the length of arc 
measured from the vertex, and the angle which the tangent line 
makes with the vertical tangent, prove that the intrinsic equation of 
the curve is ds/d$> = 2p sec 3 0, s = p sec tan +/gd~ 1 0. 

Prob. 76. The polar equation of a parabola being r = a sec 2 -2$, 
referred to its focus as pole, express s in terms of #. 

Prob. 77. Find the intrinsic equation of the curve j>/tf = cosh x/a t 
and of the curve y/a = log sec x/a. 

Prob. 78. Investigate a formula of reduction f or / cosh" .#</.#; 
also integrate by parts cosh" 1 .* Jx, tanh" 1 .*: dx t (sinh" 1 x)*dx\ and 
show that the ordinary methods of reduction for / cos wl .*sin ff .*dk: 

be applied to / cosh w x sinh" x dx. 



can 



ART. 27. FUNCTIONS OF COMPLEX NUMBERS. 

As vector quantities are of frequent occurence in Mathe- 
matical Physics ; and as the numerical measure of a vector 
in terms of a standard vector is a complex number of the 

form x-\-iy, in which x, y are real, and i stands for V- - i; it 
becomes necessary in treating of any class of functional oper- 
ations to consider the meaning of these operations when per- 
formed on such generalized numbers.* The geometrical defini- 
tions of cosh?/, sinh?/, given in Art. 7, being then no longer 
applicable, it is necessary to assign to each of the symbols 

*The use of vectors in electrical theory is shown in Bedell and Crehore's 
Alternating Currents, Chaps. XTV-XX (first published in 1892). The advantage 
of introducing the complex 'measures of such vectors into the differential equa- 
tions is shown by Steinmetz, Proc. Elec. Congress, 1893; while the additional 
convenience of expressing the solution in hyperbolic functions of these complex 
numbers is exemplified by Kennelly, Proc. American Institute Electrical 
Engineers, April 1895. (See below, Art. 37.) 



FUNCTIONS OF COMPLEX NUMBERS. 39 

cosh (x -f- iy) y sinh (x -\- iy), a suitable algebraic meaning, 
which should be consistent with the known algebraic values of 
cosh x, sinh x, and include these values as a particular case 
when/ = o. The meanings assigned should also, if possible, 
be such as to permit the addition-formulas of Art. 1 1 to be 
made general, with all the consequences that flow from them. 

Such definitions are furnished by the algebraic develop- 
ments in Art. 16, which are convergent for all values of u, real 
or complex. Thus the definitions of cosh (x -f- iy\ sinh (x -f- iy) 
are to be 



cosh (x + iy) = I + (* .+ *" + j 

I .52) 

sinh (x + iy) = (x + z + (* + * 3 + . . . 

^ j 

From these series the numerical values of cosh (x ~\- iy), 
sinh (x -f- iy) could be computed to any degree of approxima- 
tion, when x and y are given. In general the results will come 
out in the complex form* 

cosh (x -f- iy) = a -f- ib, 
sinh (x -f- iy) = c -j- zV/. 
The other functions are defined as in Art. 7, eq. (9). 

Prob. 79. Prove from these definitions that, whatever u may be, 
cosh ( u) = cosh u, sinh () = sinh #, 

</ </ . . 

-7- cosh z^ = sinh u -r-sinh = cosh u. 

du du 

^cosh mu = m* cosh mu. sinh mu = m* sinh mu.\ 
du du 

*It is to be borne in mind that the symbols cosh, sinh, here stand for alge- 
braic operators which convert one number into another; or which, in the lan- 
guage of vector-analysis, change one vector into another, by stretching and 
turning. 

f The generalized hyperbolic functions usually present themselves in Mathe- 
matical Physics as the solution of the differential equation cPty/dti* = m' 2 <p, 
where <p, m t u are complex numbers, the measures of vector quantities. (See 
Art. 37.) 



-40 HYPERBOLIC FUNCTIONS. 

ART. 28. ADDITION-THEOREMS FOR COMPLEXES. 

The addition-theorems for cosh (u -\- v\ etc., where u, v are 
complex numbers, may be derived as follows. First take u, v 
.as real numbers, then, by Art. II, 

cosh (u -\- v) = cosh u cosh v + sinh u sinh v\ 



hence I + ( + *)'+ =i + ' + .. . 



This equation is true when ?/, ^ are any real numbers. It 
must, then, be an algebraic identity. For, compare the terms 
of the rth degree in the letters 21, v on each side. Those on 

the left are ;(#-)- ^) r ; and those on the right, when collected, 

form an rth-degree function which is numerically equal to the 
former for more than r values of u when v is constant, and for 
more than r values of v when u is constant. Hence the terms 
of the rth degree on each side are algebraically identical func- 
tions of u and v.* Similarly for the terms of any other degree. 
Thus the equation above written is an algebraic identity, and 
is true for all values of ?/, v, whether real or complex. Then 
writing for each side its symbol, it follows that 

cosh (u -\- v) = cosh u cosh v -f- sinh u sinh v\ (53) 
and by changing v into v, 

cosh (// -- 7') cosh u cosh v sinh u sinh v. (54) 
In a similar manner is found 

sinh (11 v) = sinh u cosh v cosh u sinh v. (55) 

In particular, for a complex argument, 

cosh (x iy) = cosh x cosh iy sinh x sinh iy, 

sinh (;r iy) = sinh # cosh iy cosh ^r sinh iy. 

* '* If two rth-degree functions of a single variable be equal for more than r 
values of the variable, then they are equal for all values of the variable, and are 
.algebraically identical." 



FUNCTIONS OF PURE IMAGINARIES. 41 

Prob. 79. Show, by a similar process of generalization,* that if 
sin Uy cos Uy exp u \ be defined by their developments in powers of 
Uy then, whatever u may be, 

sin (u -}- v) = sin u cos v + cos u sin v, 
cos (& -{- z>) = cos ^ cos v sin # sin z>, 
exp (tf + 0) = exp u exp z>. 
Prob. 80. Prove that the following are identities: 

cosh 2 u sinh 2 u = i, 
cosh u -f- sinh # = exp #, 
cosh # sinh u = exp ( u\ 
cosh w = ^[exp u -j- exp ( )], 
sinh & = |[exp u exp( u)]. 

ART. 29. FUNCTIONS OF PURE IMAGINARIES. 
In the defining identities 

cosh u = i -) r # 2 -] r 4 + 

2! 4! 

1 , i 6 
sinh u = u A -u A u A- . . ., 

3. ^' 

o * j 

put for u the pure imaginary iy, then 

cosh zj/ = i j-y -| -y* - . . . = cos j, (57) 

* T" 

I . I 

sinh rpr = fjr -| -(ri/) 3 -| i((y) + 



=* sin /' (58) 

and, by division, tanh iy = i tan y. (59) 

* This method of generalization is sometimes called the principle of the 
" permanence of equivalence of forms." It is not, however, strictly speaking, a 
"principle," but a method; for, the validity of the generalization has to be 
demonstrated, for any particular form, by means of the principle of the alge- 
braic identity of polynomials enunciated in the preceding foot-note. (See 
Annals of Mathematics, Vol. 6, p. 8r.) 

f The symbol exp u stands for "exponential function of , " which is identi- 
cal with e* when u is real. 



HYPERBOLIC FUNCTIONS. 



These formulas serve to interchange hyperbolic and circular 
functions. The hyperbolic cosine of a pure imaginary is real, 
and the hyperbolic sine and tangent are pure imaginaries. 

The following table exhibits the variation of sinh u, cosh ?/,. 
tanh ?/, exp &, as u takes a succession of pure imaginary values. 



u 


sinh u 


cosh u 


tanh u 


exp u 








I 





I 


\iit 


.ft 


7* 


i 


7(i+0 


$in 


i 





oo i 


1 


\i-rc 


7* 


--7 


7 


7(i - 


in 


o 


i 


o 


i 


\i7t 


-.71 


-7 


i 


-7(i + 


* 


, , ^ 


o 


oo i 


i 


\in 


-.71 


7 


i 


- -7(i - i) 


2l7t 





i 





i 



* In this table .7 is written for |- 4/2, = .707 .... 
Prob. 81. Prove the following identities : 

cos y = cosh iy =.-J[exp ty + exp ( />-)], 

= - sinh iy = .[exp iy exp ( />')], 



cos y -j- / sin _y cosh /y + sinh iy = exp /v, 
cosjy / sin jv = cosh iy sinh iy = exp ( ty), 
cos ty = cosh jv, sin (y = / sinh y. 

Prob. 82. Equating the respective real and imaginary parts on- 
each side of the equation cos ny -j- / sin ny = (cos y -\- i sin y) n y 
express cos ny in powers of cos y, sin y ; and hence derive the cor- 
responding expression for cosh ny. 

Prob. 83. Show that, in the identities (57) and (58), y may be 
replaced by a general complex, and hence that 

sinh (x i\ ) = i sin (y ^ ix) t 



FUNCTIONS OF X -p- ty IN THE FORM X -}- I Y. 43 

cosh{.x ty) = cos (y T ix), 
sin (x ty) = / sinh (y ^ ix), 
cos (x /y) = cosh (y =F MC). 

Prob. 84. From the product-series for sin.* derive that for 

:sinh x : 



X*\l X' \l X~ 



nx--=x\i -,1(1 -77-7111 -.-ijpl..., 

O 



BDh**(l + 3)[l + ^?Kl + 



ART. 30. FUNCTIONS OF x-\-iy IN THE FORM 
By the addition-formulas, 

cosh (x -\- iy) cosh x cosh iy -|- sinh x sinh zjp, 
sinh (x -\- iy] = sinh x cosh y/ -j- cosh x sinh zy, 
-but cosh iy = cos jj>, sinh iy = i sin ^, 

hence cosh (x 4- zV) cosh ^ cos y 4- i sinh .* sin j/, ) 

\ I -^ / .'I ^ I f S- \ 

(60) 

sinh (.# -f- iy) = sinh ^ cos y -\-i cosh ^ sin y. ) 
Thus if cosh (x -\- iy) = a -f- z#, sinh (x -\-iy) = -f- z#, then 

^ := cosh ^ cos y, # = sinh ^ sin y, ) 

, (61) 

<; = sinh x cos 7, a = cosh # sin y. ) 

From these expressions the complex tables at the end of 
this chapter have been computed. 

Writing cosh z=Z, where z x -\- iy, Z = X-\- iY', let the 
complex numbers z, Z be represented on Argand diagrams, in 
the usual way, by the points whose coordinates are (x, y), 
'{X, Y) ; and let the point z move parallel to the jj/-axis, on a 
given line x = m, then the point Z will describe an ellipse 
whose equation, obtained by eliminating y between the equa- 
tions X = cosh m cos/, F sinh m sin y, is 

I - T 

(cosh mf (sinh m) z 

and which, as the parameter m varies, represents a series of 
confocal ellipses, the /distance) between whose foci is unity. 



44: HYPERBOLIC FUNCTIONS. 

Similarly, if the point z move parallel to the .r-axis, on a given 
line y = n, the point Z will describe an hyperbola whose equa- 
tion, obtained by eliminating the variable x from the equations- 
X= cosh x cos n, Y = sinh x sin n, is 

^ a F a 

(cos #)' (sin n)" 

and which, as the parameter n varies, represents a series of 
hyperbolas corifocal with the former series of ellipses. 

These two systems of curves, when accurately drawn at 
close intervals on the Z plane, constitute a chart of the hyper- 
bolic cosine ; and the numerical value of cosh (m + in) can be 
read off at the intersection of the ellipse whose parameter is m 
with the hyperbola whose parameter is n* A similar chart can 
be drawn for sinh (x+iy), as indicated in Prob. 85. 

Periodicity of Hyperbolic Functions. The functions sinh u 
and cosh u have the pure imaginary period 2in. For 

sinh (u + 2in) =sinh u cos 271 +i cosh u sin 27: = sinh M, 
cosh (u + 2in) =cosh u cos 2n+i sinh u sin 27r = cosh u. 
The functions sinh u and cosh u each change sign when the 
argument u is increased by the half period in. For 

sinh (u+in) =sinh u cos n+i cosh u sin n = sinh u, 
cosh (u+in) = cosh u cos n+i sinh u sin n = cosh u. 
The function tanh u has the period in. For, it follows from 
the last two identities, by dividing member by member, that 

tanh (u+in) =tanh u. 
By a similar use of the addition formulas it is shown that 

sinh (u + tyn) =i cosh u, cosh (u + ^in) =i sinh u. 
By means of these periodic, half-periodic, and quarter-periodic 
relations, the hyperbolic functions of x+iy are easily expressible 
in terms of functions of x+iy, in which y' is less than JTT. 



* Such a chart is given by Kennelly, Proc. A. I. E. E., April 1895, and is 
used by him to obtain the numerical values of cosh (x + iy). sinh (x + iy), which 
present themselves as the measures of certain vector quantities in the theory of 
alternating currents. (See Art. 37.) The chart is constructed for values of x 
and of y between o and 1.2; but it is available for all values of y, on account of 
the periodicity of the functions. 



FUNCTIONS OF X + iy IN THE FORM X+iY. 45 



The hyperbolic functions are classed in the modern function- 
theory of a complex variable as functions that are singly periodic 
with a pure imaginary period, just as the circular functions are 
singly periodic with a real period, and the elliptic functions are 
doubly periodic with both a real and a pure imaginary period. 

Multiple Values of Inverse Hyperbolic Functions. It fol- 
lows from the periodicity of the direct functions that the inverse 
functions sinh" 1 m and cosh" 1 m have each an indefinite number 
of values arranged in a series at intervals of zin. That partic- 
ular value of sinh" 1 m which has the coefficient of i not greater 
than JTT nor less than JTT is called the principal value of sinh" 1 m\ 
and that particular value of cosh" 1 m which has the coefficient of i 
not greater than it nor less than zero is called the principal value 
of cosh"^. When it is necessary to distinguish between the 
general value and the principal value the symbol of the former 
will be capitalized; thus 

Sinh"" 1 m = sinh" 1 m + 2irit, Cosh" 1 m = cosh" 1 m + 2irn, 
Tanh" 1 m = tanh" 1 m+irx, 

in which r is any integer, positive or negative. 

Complex Roots of Cubic Equations. It is well known that 
when the roots of a cubic equation are all real they are expressible 
in terms of circular functions. Analogous hyperbolic expressions 
are easily found when two of the roots are complex. Let the 
cubic, with second term removed, be written 



Consider first the case in which b has the positive sign. Let 
x=r sinh u, substitute, and divide by r 3 , then 

3& 2C 

sinh U + -? smh u = -s. 

V" 4 

Comparison with the formula sinh 3 w + J sinh u = J sinh $u 

36 3 2C sinh 

gives r 

2 

whence 
therefore 



r=2&*, sinh3w=-, u = 



=2& sinh sinh~ 1 




46 HYPERBOLIC FUNCTIONS. 

in which the. sign of b* is to be taken the same a's the sign of c. 
Now let the principal value of sinn" 1 ^-, found from the tables, 
be n\ then two of the imaginary values are w2wr, hence the 



i e ti i n i f n 2 \ 

three values of x are 20* smh and 20* sinni 4 -). The 

3 \3 " 3 7 

last two reduce to 0*(sinh A/3 cosh --). 

V 3 37 

Next, let the coefficient of x be negative and equal to 30. 
It may then be shown similarly that the substitution x=r sin 
leads to the three solutions 

n n/ n , /- n \ i c 

20* sin , o*( sin v 3 cos ), where n = sin" 1 TT . 

3 V 3 37 & 

These roots are all real when c^b*. If c>$, the substitution 
x = r cosh u leads to the solution 

x = 20* cosh ( cosh" 1 77 ) * 
\3 #J 

which gives the three roots 

M / 11 tl\ f 

20* cosh, -0*( cosh *\/J sinh J, wherein n cosh" 1 jj, 
in which the sign of b* is to be taken the same as the sign of c. 

Prob. 85. Show that the chart of cosh (x -\- ty) can be adapted 
to sinh (x -j- (y), by turning through a right angle; also to sin (x ~{-ty). 

sinh 2X 4- /"sin 2V 

Prob. 86. Prove the identity tanh (x + ty) = - -. 

cosh 2x -j- cos 2y 

Prob. 87. If cosh (x + /v), = a -f- /^, be written in the " modulus 
and amplitude" form as r(cos -f- /'sin 0), = r exp /#, then 

r a = a? -j- ^ a = cosh 2 x sin a 7 = cos 2 ^ sinh 8 x, 

tan = b/a = tanh ar tan y. 
Prob. 88. Find the modulus and amplitude of sinh (x -j- ty). 



Prob. 80. Show that the period of exp - is ia. 



a 



Prob. 90. When m is real and > i, cos" 1 m = i cosh" 1 m t 

1 m = -- /cosh" 1 m. 
2 

When m is real and < i, cosh" 1 m = /cos" 1 m. 



THE CATENARY. 47 

ART. 31. THE CATENARY. 

A flexible inextensible string is suspended from two fixed 
points, and takes up a position of equilibrium under the 
action of gravity. It is required to find the equation of the 
curve in which it hangs. 

Let w be the weight of unit length, and s the length of arc 
AP measured from the lowest point A ; then ws is the weight 
of the portion AP. This is balanced by the terminal tensions, 
T acting in the tangent line at P, and H in the horizontal 
tangent. Resolving horizontally and vertically gives 

T cos = H) T sin = ws, 
in which is the inclination of the tangent at P\ hence 

ws s 



where c is written for H/w, the length whose weight is the 
constant horizontal tension ; therefore 



dv s ds / s dx ds 

-7 = -. -J = <i / * + T ~~ ~7== 

dx c dx \l c c t/V _|_ 8 ' 

^ 5 ^rj^/J # 

= sinh- 1 , sinh = -=, = cosh -, 
c c c c ax t- c 

which is the required equation of the catenary, referred to an 
axis of x drawn at a distance c below A. 

The following trigonometric method illustrates the use of 
the gudermanian : The " intrinsic equation," s = c tan 0, 
gives ds = c sec 2 d<k> ; hence dx, = ds cos 0, = c sec d<p ; 
dy, ds sin 0, = <;sec tan 0^0 ; thus xc gd -1 0, jj/ = sec 0; 
whence y/c = sec = sec gd ;tr/ = cosh x/c ; and 
,y/ = tan gd x/c = sinh x/c. 

Numerical Exercise. A chain whose length is 30 feet is 
suspended from two points 20 feet apart in the same hori- 
zontal ; find the parameter c, and the depth of the lowest 
point. 



48 HYPERBOLIC FUNCTIONS. 

The equation s/c = sinh x/c gives i$/c = sinh lo/c, which, 
by putting lo/c = z t may be written 1.5,2 = sinh .2. By exam- 
ining the intersection of the graphs of/ = sinh#, y = 1.5.2, 
it appears that the root of this equation is z = 1.6, nearly. 
To find a closer approximation to the root, write the equation 
in the form/(V) = sinh z 1.5,2 = o, then, by the tables, 

/(i.6o) = 2.3756 -- 2.4000 = .0244, 
f(i.62) = 2.4276 2.4300 = - - .0024, 
/(i.64) = 2.4806 -- 2.4600 = -f- .0206; 

whence, by interpolation, it is found that f(i.622i) = o, and 
z = 1.6221, 10/2 = 6.1649. The ordinate of either of 
the fixed points is given by the equation 

y/c cosh x/c = cosh lO/c = cosh 1.6221 = 2.6306, 

from tables; hence y =. 16.2174, and required depth of the 
vertex = y c = 10.0525 feet.* 

Prob. 91. In the above numerical problem, find the inclination 
of the terminal tangent to the horizon. 

Prob. 92. If a perpendicular MN be drawn from the foot of the 
ordinate to the tangent at P, prove that MN is equal to the con- 
stant c, and that NP is equal to the arc AP. Hence show that 
the locus of N is the involute of the catenary, and has the prop- 
erty that the length of the tangent, from the point of contact to the 
axis of x, is constant. (This is the characteristic property of the 
tractory). 

Prob. 93. The tension T at any point is equal to the weight of a 
portion of the string whose length is equal to the ordinate y of that 
point. 

Prob. 94. An arch in the form of an inverted catenary f is 30 
feet wide and 10 feet high; show that the length of the arch can be 

2 3O 

obtained from the equations cosh z z = i, 2S = sinh z. 

3 z 

* See a similar problem in Chap. I, Art. 7. 

f For the theory of this form of arch, see " Arch " in the Encyclopaedia 
Britannica. 



CATENARY OF UNIFORM STRENGTH. 49 

ART. 32. CATENARY OF UNIFORM STRENGTH. 

If the area of the normal section at any point be made 
proportional to the tension at that point, there will then be a 
constant tension per unit of area, and the tendency to break 
will be the same at all points. To find the equation of the 
curve of equilibrium under gravity, consider the equilibrium of 
an element PP' whose length is ds, and whose weight is gpoads, 
where GO is the section at P, and p the uniform density. This 
weight is balanced by the difference of the vertical components 
of the tensions at P and P', hence 

d( T sin 0) = gpcods, 
d(T cos 0) = o ; 

therefore T cos H, the tension at the lowest point, and 
T = H sec 0. Again, if G? O be the section at the lowest point, 
then by hypothesis OO/GO O = T/H = sec 0, and the first equation 
becomes 

Hd(szc sin 0) gpco sec ds, 

or cdtan = 



where c stands for the constant H/gpoo^ the length of string 
(of section o? ) whose weight is equal to the tension at the 
lowest point ; hence, 

ds = c sec 0^0, s/c = gd -1 0, 
the intrinsic equation of the catenary of uniform strength. 

Also dx = ds cos = c dcf), dy = ds sin = c tan dcf> ; 
hence x c<p, y = c log sec 0, 

and thus the Cartesian equation is 

y/c = log sec x/c, 
in which the axis of x is the tangent at the lowest point. 

Prob. 95. Using the same data as in Art. 3i find the parameter 
c and the depth of the lowest point. (The equation x/c = gd s/c 
gives lo/c = gd I5/V, which, by putting i$/c = z, becomes 



50 HYPERBOLIC FUNCTIONS. 

gd z = \z. From the graph it is seen that z is nearly 1.8. If 
f(z) = gd z \z y then, from the tables of the gudermanian at the 
end of this chapter, 

/(i.8o) = 1.2432 1.2000 = -f- .0432, 
/(i. 9 o) - 1.2739 1-2667 = + .0072, 
/( r '9S) 1-2881 1.3000 = .0119, 

whence, by interpolation, z =.1.9189 and =7.8170. Again, 
y/c = loge sec x/c ; but x/c = IQ/C = 1.2793; and 1.2793 radians 
= 73 17' 55"; hence 7 = 7.8170 X -4I9H X 2.3026 = 7.5443* the 
required depth.) 

Prob. 96. Find the inclination of the terminal tangent. 

Prob. 97. Show that the curve has two vertical asymptotes. 

Prob. 98. Prove that the law of the tension T, and of the section 
GO, at a distance s, measured from the lowest point along the 
curve, is 

T GO S 

= = cosh -; 
H GO O c j 

and show that in the above numerical example the terminal section 
is 3.48 times the minimum section. 

Prob. 99. Prove that the radius of curvature is given by 
p = c cosh s/c. Also that the weight of the arc s is given by 
W H sinh s/c t in which s is measured from the vertex. 

ART. 33, THE ELASTIC CATENARY. 

An elastic string of uniform section and density in its natu- 
ral state is suspended from two points. Find its equation of 
equilibrium. 

Let the element do- stretch"into ds\ then, by Hooke's law, 
ds da(\ -\- AT"), where A is the elastic constant of the string ; 
hence the weight of the stretched element ds, = gpcadv, = 
,gpGods/(i + AT). Accordingly, as before, 

d(Tsm 0) =gpa>ds/(i + AT), 
and T cos = H = gpooc, 

hence cd(tan 0) ds/(i + // sec 0), 

in which jj. stands for A//, the extension at the lowest point ; 



THE TRACTORY. 



51 



therefore ds = <:(sec 2 + ^ sec3 

s/c = tan + ju( sec tan + gd" 1 0), [prob. 20, p. 37 

which is the intrinsic equation of the curve, and reduces to that 
of the common catenary when JLI o. The coordinates x, y 
may be expressed in terms of the single parameter by put- 
ting dx = ds cos = <:(sec -f- /* sec 2 0)^0, 

dy = ds sin = ^(sec 2 -j~ yw sec 3 0) sin </0. Whence 



= gd" 1 -f- /* tan 0, 



sec 



0- 



These equations are more convenient than the result of 
eliminating 0, which is somewhat complicated. 

/ART. 34. THE TRACTORY.* 

To find the equation of the curve which possesses the 
property that the length of the tangent from the point of con- 
tact to the axis of x is con- 
stant. 

Let PT, P'T' be two con- 
secutive tangents such that 
PT=P'T' = c, and let OT 
= t\ draw TS perpendicular 
to/" 7*'; then if PP' = ds, it 
is evident that ST' differs 
from ds by an infinitesimal of a higher order. Let PT make 
an angle with OA, the axis of y\ then (to the first order of 
infinitesimals) PTdcf) = TS = TT' cos 0; that is, 

ctfcf) = cos 0<//, / = c gd~ J 0, 

x = / c sin 0, c(gd~* sin 0), y = c cos 0. 
This is a convenient single-parameter form, which gives all 

*This curve is used in Schiele's anti-friction pivot (Minchin's Statics, Vol. I, 
p. 242) ; and in the theory of the skew circular arch, the horizontal projection 
of the joints being a tractory. (See "Arch," Encyclopaedia Britannica.) The 
equation (p = gd t/c furnishes a convenient method of plotting the curve. 




52 HYPERBOLIC FUNCTIONS. 

values of x, y as increases from o to \n. The value of s, ex- 
pressed in the same form, is found from the relation 

ds = ST f dt sin = c tan (pd<p, s = c log, sec 0. 

At the point A, o, x = o, s = o, / = o, y=c. The 
Cartesian equation, obtained by eliminating 0, is 



= gd" 1 (cos" 1 -] sin (cos~ ! -\ cosh" 1 \ / 1 ?- 

c \ / \ c] y \ <? 

If & be put for //, and be taken as independent variable, 
= gd u, x/c = u tanh.^, y/c sech u, s/c = log cosh ?/. 

Prob. 100. Given / = 2C, show that = 74 35', j = 1.3249*:, 
^ := .2658*:, # = 1.0360*:. At what point is t = cl 

Prob. 101. Show that the evolute of the tractory is the catenary. 
(See Prob. 92.) 

Prob. 102. Find the radius of curvature of the tractory in terms 
of ; and derive the intrinsic equation of the involute. 

V 

^ ART. 35. THE LOXODROME. 

On the surface of a sphere a curve starts from the equator 
in a given direction and cuts all the meridians at the same 

angle. To find its equation 
in latitude-and-longitude co- 
ordinates : 

Let the loxodrome cross 
two consecutive meridians 
AM, AN in the points/*, Q\ 
let PR be a parallel of lati- 
tude ; let OM= x, MP=.y, 
MN = dx, RQ dy, all in radian measure ; and let the angle 
MOP=RPQ = a\ then 

tan a = RQ/PR, but PR = MN cos MP* 

hence dx tan a = dy sec y, and x tan a = gd' 1 y, there being 
no integration-constant since y vanishes with x ; thus the re- 
quired equation is 

y = gd (x tan a). 

* Jones, Trigonometry (Ithaca, 1890), p. 185. 




COMBINED FLEXURE AND TENSION. 53 

To find the length of the arc OP: Integrate the equation 
ds = dy esc a, whence s = y esc <x. 

To illustrate numerically, suppose a ship sails northeast, 
from a point on the equator, until her difference of longitude is 
45, find her latitude and distance : 

Here tan a = I, andjj> = gd x = gd \n = gd (.7854) = .7152 
radians: s = y\/2 = 1.0114 radii. The latitude in degrees is 
40.980. 

If the ship set out from latitude y lt the formula must be 
modified as follows : Integrating the above differential equa- 
tion between the limits (#,, y t ) and (x^y^ gives 

(x, - O tan a = gd-> 2 - gd->,; 

hence gd~'j/ 2 = gd" 1 ^ + (x* x^) tan a, from which the final 
latitude can be found when the initial latitude and the differ- 
ence of longitude are given. The distance sailed is equal to 
{y* ~" Ji) csc a radii, a radius being 60 X i8o/?r nautical miles. 
Mercator's Chart. In this projection the meridians are 
parallel straight lines, and the loxodrome becomes the straight 
line y' = x tan a, hence the relations between the coordinates of 
corresponding points on the plane and sphere are x' = x, 
y' = gd~ l y. Thus the latitude y is magnified into gd ~ l y, which 
is tabulated under the name of "meridional part for latitude 
y " ; the values of y and of y' being given in minutes. A chart 
constructed accurately from the tables can be used to furnish 
graphical solutions of problems like the one proposed above. 

Prob. 103. Find the distance on a rhumb line between the points 
(30 N, 20 E) and (30 S, 40 E). 

ART. 36. COMBINED FLEXURE AND TENSION. 

A beam that is built-in at one end carries a load P at the 
other, and is also subjected to a horizontal tensile force Q ap- 
plied at the same point; to find the equation of the curve 
assumed by its neutral surface : Let x> y be any point of the 



54 HYPERBOLIC FUNCTIONS. 

elastic curve, referred to the free end as origin, then the bend- 
ing moment for this point is Qy Px. Hence, with the usual 
notation of the theory of flexure,* 



which, on putting^ mx = #, a&dd*y/dx* =zd*u/dx*, becomes 



= tfu, 
dx 

whence u = A cosh nx -f- B sinh #, [probs. 28, 30 

that is, y = mx ~\- A cosh nx + B sinh ;r. 

The arbitrary constants A, B are to be determined by the 
terminal conditions. At the free end x = o, y = o ; hence A 
must be zero, and 

y = mx -\- B sinh nx, 

-f- m -{- nB cosh nx ; 
dx 

but at the fixed end, x = /, and dy/dx = o, hence 

B = m/n cosh /, 
and accordingly 

w sinh nx 

y =. mx -- -: 7. 
n cosh nl 

To obtain the deflection of the loaded end, find the ordinate 
of the fixed end by putting x = /, giving 

deflection = m(l -- tanh nl\ 

n ' 

Prob. 104. Compute the deflection of a cast-iron beam, 2X2 
inches section, and 6 feet span, built-in at one end and carrying 
a load of 100 pounds at the other end, the beam being subjected 
to a horizontal tension of 8000 pounds. [In this case / = 4/3, 
E = 15 X io 6 , Q = 8000, P = 100 ; hence n = 1/50, m = 1/80, 
deflection = ^(72 50 tanh 1.44) = -gV(7 2 44-69) = -34i inches.] 

* Merriman, Mechanics of Materials (New York, 1895), pp. 70-77, 267-269. 



ALTERNATING CURRENTS. 55' 

Prob. 105. If the load be uniformly distributed over the beam r 
say w per linear unit, prove that the differential equation is 



73 

or ~ = 



2JH 

and that the solution isy = A cosh # + ^? sinh nx-\-mx iJ \ ^. 

n 

Show also how to determine the arbitrary constants. 



ART. 37. ALTERNATING CURRENTS.* 

In the general problem treated the cable or wire is regarded 
as having resistance, distributed capacity, self-induction, and 
leakage ; although some of these may be zero in special 
cases. The line will also be considered to feed into a receiver 
circuit of any description ; and the general solution will in- 
clude the particular cases in which the receiving end is either 
grounded or insulated. The electromotive force may, without 
loss of generality, be taken as a simple harmonic function of 
the time, because any periodic function can be expressed in a 
Fourier series of simple harmonics.f The E.M.F. and the 
current, which may differ in phase by any angle, will be 
supposed to have given values at the terminals of the receiver 
circuit ; and the problem then is to determine the E.M.F, 
and current that must be kept up at the generator terminals ; 
and also to express the values of these quantities at any inter- 
mediate point, distant x from the receiving end ; the four 
line-constants being supposed known, viz.: 

R = resistance, in ohms per mile, 

L = coefficient of self-induction, in henrys per mile, 

C = capacity, in farads per mile, 

G = coefficient of leakage, in mhos per mile.J 

It is shown in standard works that if any simple harmonic 

* See references in foot-note Art. 27. f Chapter V, Art. 8. 

\ Kennelly denotes these constants by r, /, c, g. Steinmetz writes s for 
aoLy K for ooC, Q for G, and he uses C*for current. 

Thomson and Tait, Natural Philosophy, Vol, I. p. 40; Rayleigh, Theory 
of Sound, Vol. I. p. 20; Bedell and Crehore, Alternating Currents, p. 214. 



56 HYPERBOLIC FUNCTIONS. 

function a sin (cot -)- 6) be represented by a vector of length 
a and angle 6, then two simple harmonics of the same period 
27T/C0, but having different values of the phase-angle 6, can be 
combined by adding their representative vectors. Now the 
E.M.F. and the current at any point of the circuit, distant x 
from the receiving end, are of the form 

e = e l sin (cot -f- 0), i = i l sin (c&t + 0'), (64) 

in which the maximum values e lt t lt and the phase-angles 0, 6', 
are all functions of x. These simple harmonics will be repre- 
sented by the vectors *,/0, iJO' ; whose numerical measures 
are the complexes *, (cos 6 -\-jsin &)*, z, (cos 6' -\- j sin 0'), 
which will be denoted by e, L The relations between e and z 
may be obtained from the ordinary equations f 

di de de di 



for, since de/dt = coe } cos (&tf -{- 0) = ooe l sin (&?/ -f- 6 -j- |-TT), 
then dk/dT/ will be represented by the vector coe 1 /0 -\-^n ; and 



di/dx by the sum of the two vectors Ge l /0, Ccoe^ /& + i 7r 5 
whose numerical measures are the complexes Ge,jooCe; and 
similarly for de/dx in the second equation ; thus the relations 
between the complexes /, F are 



*In electrical theory the symbol j is used, instead of z, for |/ I. 

f Bedell and Crehore, Alternating Currents, p. 181. The sign of dx is 
changed, because x is measured from the receiving end. The coefficient of 
leakage, G, is usually taken zero, but is here retained for generality and sym- 
metry. 

^ These relations have the advantage of not involving the time. Steinmetz 
derives them from first principles without using the variable t. For instance, 
he regards R -\-jooL as a generalized resistance-coefficient, which, when applied 
to i, gives an E.M.F., part of which is in phase with i, and part in quadrature 
with i. Kennelly calls R -\- j'ooL the conductor impedance; and G -f- jc&C the 
dielectric admittance; the reciprocal of which is the dielectric impedance. 



ALTERNATING CURRENTS. 57 

Differentiating and substituting give 



(67) 



and thus e, i are similar functions of x, to be distinguished only 
by their terminal values. 

It is now convenient to define two constants ;/z, m l by the 
equations* 

m* = (R +ja>L)(G +jot>Q 9 m, = m/(G +jot>C) ; (68) 
and the differential equations may then be written 



nri > (69) 

the solutions of which are f 

e = A cosh mx -f- B sinh mx, i = A' cosh mx -\- B' sinh mx, 

wherein only two of the four constants are arbitrary ; for sub- 
stituting in either of the equations (66), and equating coeffi- 
cients, give 



(G+jct>C)A 

whence B' = A/m lt A' B/m,. 

Next let the assigned terminal values of e, z, at the receiver, 
be denoted by E, /; then puttings = o gives E = A, f= A', 
whence B = mj, B' = E/m l ; and thus the general solution is 

e = E cosh mx -\- mj sinh mx, 

'. (70) 

z = / cosh mx -j- m E sinh mx. 

* The complex constants m, m\ , are written z, y by Kennelly; and the 
variable length x is written Z 2 . Steinmetz writes v for m. 
\ See Art. 14, Probs 28-30; and Art. 27, foot-note. 



58 HYPERBOLIC FUNCTIONS. 

If desired, these expressions could be thrown into the ordi- 
nary complex form X -\-jY, X' -\-jY r , by putting for the let- 
ters their complex values, and applying the addition-theorems- 
for the hyperbolic sine and cosine. The quantities X, Y, X 1 ', 
Y' would then be expressed as functions of x ; and the repre 
sentative vectors of e, i y would be e^/6, i l /&', where e* = X 3 -|- F 2 , 
*; = X" + Y'\ tan 6 = Y/X, 



For purposes of numerical computation, however, the for- 
mulas (70) are the most convenient, when either a chart,* or a; 
table,f of cosh &, sinh u, is available, for complex values of u.. 

Prob. io6.J Given the four line-constants: R= 2 ohms per mile, 
L = 20 millihenrys per mile, C= 1/2 microfarad per mile, G = o> 
and given GO, the angular velocity of E.M.F. to be 2000 radians, 
per second; then 

ooL = 40 ohms, conductor reactance per mile; 
It -{-j'ooL = 2 -f- 407 ohms, conductor impedance per mile; 

coC = .001 mho, dielectric susceptance per mile; 
G -\-j(&C = .00 1/ mho, dielectric admittance per mile; 
(G-\-jooC)~ l = looq/ohms, dielectric impedance per mile; 

m * = (R-\-jooL}(G -\-jGoC) .04 + .0027, which is the^ 
measure of .04005 /i77 8'; therefore 

m = measure of .2001 /88 34' = .0050 -f- .2000;', an ab- 
stract coefficient per mile, of dimensions [length]' 1 ,. 
m^ m/(G -\-jcoC) = 200 5; ohms. 

Next let the assigned terminal conditions at the receiver be:: 
/= o (line insulated); and E = 1000 volts, whose phase may be taken, 
as the standard (or zero) phase; then at any distance x, by (70),, 

E 

e = E cosh nix. i -= sinh mx, 

m, 

in which mx is an abstract complex. 

Suppose it is required to find the E.M.F. and current that must 
be kept up at a generator 100 miles away; then 

* Art. 30, foot-note. f See Table II. 

J The data for this example are taken from Kennelly's article (1. c., 
P- 38). 



ALTERNATING CURRENTS. 59 

e = 1000 cosh (.5 -|- 207), i = 200(40 7)"' sinh (.5 + 2O /) 
ibut, by Prob. 89, cosh (.5 + 207 ) = cosh (.5 + 207 67T/) 

= cosh (.5 -h 1.15;) = .4600 + .475/ 

obtained from Table II, by interpolation between cosh (.5 + iy) 
and cosh (.5 + I - 2 /)l hence 

e = 460 + 475;' = e^cos 6 +7 sin 6), 

where log tan log 475 log 460 = .0139, = 45 55', and 
l = 460 sec 9 = 661.2 volts, the required E.M.F. 

Similarly sinh (.5 + 207) = sinh (.5 -j- 1.157) .2126+1.0280;', 
and hence 



+ /)(- 2126 + 1.028;) = 7( T 495 + 8266;') 

= /;(cos 0'+7'sin 6>'), 

where log tan 6' = 10.7427, 0' = 79 45', *', = 1495 sec 
5.25 amperes, the phase and magnitude of required current. 

Next let it be required to find e at x = 8; then 

e= 1000 cosh (.04 + i.6/) = iooo/ sinh (.04+ .037), 

by subtracting ^TT/, and applying page 44. Interpolation be 
tween sinh (o + o/) and sinh (o + .i/) gives 

sinh (o + .037) = ooooo + . 029957. 
Similarly sinh (.1 -f -037) = .10004 + -0300471 

Interpolation between the last two gives 

sinh (.04 + .037) = .04002 + .O2999/. 

Hence ^=7(40.02 +29.997')= 29.99+40.027 =e l (cos $+7" sin #), 
-where 



log tan = .12530, = 126 51',*?, = 29.99 sec I2 ^ 5 1 ' = 5- 01 
volts. 

Again, let it be required to find e at x = 16; here 

e = iooo cosh (.08 + 3.27) = iooo cosh (.08 + .067'), 
l)ut cosh (o + .067) = .9970 + 07", cosh (.1 + .067) = 1.0020 + .0067; 
hence cosh (.08 + .067)^:1.0010 +.00487', 

and e= 1001+4.87 = ^(cos #+7' sin 0), 

where 6 = 180 17', e l = 1001 volts. Thus at a distance of about 
16 miles the E.M.F. is the same as at the receiver, but in opposite 



60 HYPERBOLIC FUNCTIONS. 

phase. Since e is proportional to cosh (.005 -j- .2j)x, the value of 
x for which the phase is exactly 180 is n/.2 15.7. Similarly 
the phase of the E.M.F. at x 7.85 is 90. There is agreement 
in phase at any two points whose distance apart is 31.4 miles. 

In conclusion take the more general terminal conditions in 
which the line feeds into a receiver circuit, and suppose the current 
is to be kept at 50 amperes, in a phase 40 in advance of the elec- 
tromotive force; then / 5o(cos 40 -\-j sin 40) = 38.30 -j- 32.147, 
and substituting the constants in (70) gives 

<? = 1000 cosh (.005 -f- .2/)x -f (7821 + 62367) sinh (.005 -f .2j}x 
= 460+ 4757 -4748+93667= -4288+98417 = <? x (cos 0+/sin 0),. 

where 0= 113 33', e l = 10730 volts, the E.M.F. at sending end. 

This is 17 times what was required when the other end was insulated. 

Prob. 107. If L = o, G = o, / = o ; then m = (i -f- j)n r 

m, = (i +./),, where n 2 aoRC/2, n? = R/zGoC; and the solution is 

<?, = ~~p-E /cosh 2nx -)- cos 2tix, tan = tan nx tanh nx, 

V 2 

i. = - E I/cosh 2nx cos 2nx> tan 6' = tan nx coth nx. 

2, 

Prob. 108. If self-induction and capacity be zero, arid the 
receiving end be insulated, show that the graph of the electromotive 
force is a catenary if G ^ o, a line if G = o. 

Prob. 109. Neglecting leakage and capacity, prove that the 
solution of equations (66) is i = 7, e = -\- (fi ~\-JGoL)Ix. 

Prob. no. If x be measured from the sending end, show how 

equations (65), (66) are to be modified; and prove that 

_ i _ 

e = jE^ cosh mx m 1 7 sinh mx, i = 7 cosh mx ~ sinh mx t . 

where / refer to the sending end. 

ART. 38. MISCELLANEOUS APPLICATIONS. 

1. The length of the arc of the logarithmic curve y a* is 
s = M(cos\\ ?/+logtanh %u\ in which M= i/log a, sinh u y/M. 

2. The length of arc of the spiral of Archimedes r = a6 is. 
s = <2(sinh 2u -j- 2u), where sinh u = 6. 

3. In the hyperbola x*/a* y* /& = I the radius of curva- 
ture is p (a* sinh 2 u -f- tf cosh 2 u)l/ab ; in which u is the 
measure of the sector AOP, i.e. cosh u x/a, sinh it =y/b. 

4. In an oblate spheroid, the superficial area of the zone 



MISCELLANEOUS APPLICATIONS. 61 

between the equator and a parallel plane at a distance^/ is 
5 = 7fd\smh 2n -f- 2u)/2e, wherein b is the axial radius, e eccen- 
tricity, sinh u ey/p, and/ parameter of generating ellipse. 

5. The length of the arc of the parabola y = 2px, measured 
from the vertex of the curve, is /==^(sinh 211 -\-2ii), in which 
sinh u y/p tan 0, where is the inclination of the termi- 
nal tangent to the initial one. 

6. The centre of gravity of this arc is given by 

3/^r =r/ 2 (cosh 3 u - - i), 6^ly / 2 (sinh 4u 421} ; 

and the surface of a paraboloid of revolution is = 27t yl. 

7. The moment of inertia of the same arc about its ter. 

minal ordinate is / = ^\xl(x 2x] + ^p 3 N~\, where /< is 
the mass of unit length, and 

// = u ^ sinh 2u \ sinh 4&-f- T V sinh 6u. 

8. The centre of gravity of the arc of a catenary measured 
from the lowest point is given by 

4/y= 2 (sinh 2u + 2u), lx '= c*(u sinh u cosh u -f- i), 

in which u =x/c\ and the moment of inertia of this arc about 
its terminal abscissa is 

/ = /^Xir s ' n h 3^ H~ f sm h u u cos h u )- 

9. Applications to the vibrations of bars are given in Ray- 
leigh, Theory of Sound, Vol. I, art. 170; to the torsion of 
prisms in Love, Elasticity, pp. 166-74; to the flow of heat 
and electricity in Byerly, Fourier Series, pp. 75-81 ; to wave 
motion in fluids in Rayleigh, Vol. I, Appendix, p. 477, and in 
Bassett, Hydrodynamics, arts. 120, 384; to the theory of 
potential in Byerly p. 135, and in Maxwell, Electricity, arts. 
172-4; to Non-Euclidian geometry and many other subjects 
in Gunther, Hyperbelfunktionen, Chaps. V and VI. Several 
numerical examples are worked out in Laisant, Essai sur les 
fonctions hyperboliques. 



.0.^ HYPERBOLIC FUNCTIONS. 

ART. 39. EXPLANATION OF TABLES. 

In Table I the numerical values of the hyperbolic functions 
sinh w, cosh u, tanh u are tabulated for values of u increasing 
from o to 4 at intervals of .02. When u exceeds 4, Table IV 
may be used. 

Table II gives hyperbolic functions of complex arguments, 
in which 

cosh (x iy) = a _ ib, sinh (x iy) = c id, 

.and the values of a, b, c, d are tabulated for values of x 
.and of y ranging separately from o to 1.5 at intervals of .1. 
When interpolation is necessary it may be performed in three 
stages. For example, to find cosh (.82 -f- 1.342) : First find 
cosh (.82 -f- i-30 by keeping^ at i-3 and interpolating between 
the entries under x = .8 and x = .9 ; next find cosh (.82 -f- i-4*') 
by keeping^ at 1.4 and interpolating between the entries under 
x = .8 and x = .9, as before; then by interpolation between 
cosh (.82-)- 1.33) and cosh (.82 + 1-40 find cosh( .82 + i-34*) 
in which x is kept at .82. The table is available for all values 
of y, however great, by means of the formulas on page 44: 

sinh (x -f- 2i7i ) = sinh^r, cosh (x-\- 2i7t) = cosh x, etc. 

Jt does not apply when x is greater than 1.5, but this case sel- 
dom occurs in practice. This table can also be used as a com- 
plex table of circular functions, for 

cos (y ix] = a q= ib, sin (y ix) = d ic ; 
.and, moreover, the exponential function is given by 
.exp ( x iy) = a c i(b d), 

in which the signs of c and d are to be taken the same as the 
sign of X, and the sign of i on the right is to be the product of 
the signs of x and of i on the left. 

Table III gives the values of v= gd u, and of the guder- 
jnanian angle 9 = j8o v/n^ as u changes from o to I at inter- 



EXPLANATION OF TABLES. 63 

vals of .02, from I to 2 at intervals of .05, and from 2 to 4 at 
intervals of .1. 

In Table IV are given the values of gd u, log sinh u, log 
cosh u, as u increases from 4 to 6 at intervals of .1, from 6 to 
7 at Intervals of .2, and from 7 to 9 at intervals of .5. 

In the rare cases in which more extensive tables are neces- 
sary, reference may be made to the tables* of Gudermann, 
Glaisher, and Geipel and Kilgour. In the first the Guderman- 
ian angle (written k) is taken as the independent variable, and 
increases from o to 100 grades at intervals of .01, the corre- 
sponding value of u (written Lk] being tabulated. In the usual 
case, in which the table is entered with the value of &, it gives 
by interpolation the value of the gudermanian angle, whose 
circular functions would then give the hyperbolic functions 
of u. When u is large, this angle is so nearly right that inter- 
polation is not reliable. To remedy this inconvenience Gu- 
dermann's second table gives directly log sinh u, log cosh u, 
log tanh u, to nine figures, for values of u varying by .001 from 
2 to 5, and by .01 from 5 to 12. 

Glaisher has tabulated the values of e* and *-*, to nine sig- 
nificant figures, as x varies by .001 from o to .1, by .01 from o 
to 2, by .1 from o to 10, and by I from o to 500. From these 
the values of cosh x, sinh x are easily obtained. 

Geipel and Kilgour's handbook gives the values of cosh^, 
sinh x, to seven figures, as x varies by .01 from o to 4. 

There are also extensive tables by Forti, Gronau, Vassal, 
Callet, and Hoiiel ; and there are four-place tables in Byerly's 
Fourier Series, and in Wheeler's Trigonometry. 

In the following tables a dash over a final digit indicates 
that the number has been increased. 

* Gudermann in Crelle's Journal, vols. 6-9, 1831-2 (published separately 
under the title Theorie der hyperbolischen Functionen, Berlin, 1833). Glaisher 
in Cambridge Phil. Trans., vol. 13, 1881. Geipel and Kilgour's Electrical Hand- 
book. 



64 



HYPERBOLIC FUNCTIONS. 



TABLE I. HYPERBOLIC FUNCTIONS. 



. 


sinh u. 


cosh u. 


tanh . 


u. 


sinh u. 


cosh u. 


tanh u. 


.00 


.0000 


1.0000 


.0000 


1.00 


1.1752 


1.5431 


.7616 


02 


0200 


1.0002 


0200 


1.02 


1.2063 


1.5669 


7699 


04 


0400 


1.0008 


0400 


1.04 


1.2379 


1.5913 


7779 


06 


0600 


1.0018 


0599 


1.06 


1.2700 


1.6164 


7857 


08 


0801 


1.0032 


0798 


1.08 


1.3025 


1.6421 


7932 


.10 


.1002- 


1.0050 


.0997 


1.10 


1.3356 


1.6685 


.8005 


12 


1203 


1 0072 


1194 


1.12 


1.3693 


1.6956 


8076 


14 


1405 


1.0098 


1391 


1.14 


1.4035 


1.7233 


8144 


16 


1607 


1.0128 


1586 


1.16 


1.4382 


1.7517 


8210 


18 


1810 


1.0162 


1781 


1.18 


1.4735 


1.7808 


8275 


.20 


.2013 


1.0201 


.1974 


1.20 


1.5095 


1.8107 


.8337 


22 


2218 


1.0243 


2165 


1.22 


1.5460 


1.8412 


8397 


24 


2423 


1.0289 


2355 


1.24 


1.5831 


1.8725 


8455 


26 


2629 


1.0340 


2543 


1.26 


1.6209 


1.9045 


8511 


28 


2837 


1.0395 


2729 


1.28 


1.6593 


1.9373 


8565 


.30 


.3045 


1.0453 


.2913 


1.30 


1.6984 


1.9709 


.8617 


32 


3255 


1.0516 


3095 


1.32 


1.7381 


2.0053 


8668 


34 


3466 


1.0584 


3275 


1.34 


1.7786 


2.0404 


8717 


36 


3678 


1.0655 


3452 


1.36 


1.8198 


2.0764 


8764 


38 


3892 


1.0731 


3627 


1.38 


1.8617 


2.1132 


8810 


.40 


.4108 


1.0811 


.3799 


1.40 


1.9043 


2.1509 


.8854 


42 


4325 


1.0895 


3969 


1.42 


1.9477 


2.1894 


8896 


44 


4543 


1.0984 


4136 


1.44 


1.9919 


2.2288 


8937 


46 


4764 


1.1077 


4301 


1.46 


2.0369 


2.2691 


8977 


48 


4986 


1.1174 


4462 


1.48 


2.0827 


2.3103 


9015 


.50 


.5211 


1.1276 


.4621 


1.50 


2.1293 


2.3524 


.9051 


52 


5438 


1.1383 


4777 


1.52 


2.1768 


2.3955 


9087 


54 


5666 


1.1494 


4930 


1.54 


2.2251 


2.4395 


9121 


56 


5897 


1.1609 


5080 


-1.56 


2.2743 


2.4845 


9154 


58 


6131 


1.1730 


5227 


1.58 


2.3245 


2.5305 


9186 


.60 


.6367 


1.1855 


.5370 


1.60 


2.3756 


2.5775 


.9217 


62 


6605 


1.1984 


5511 


1.62 


2.4276 


2.6255 


9246 


64 


6846 


1.2119 


5649 


1.64 


2.4806 


2.6746 


9275 


66 


7090 


1.2258 


5784 


1.66 


2.5346 


2.7247 


9302 


68 


7336 


1.2402 


5915 


1.68 


2.5896 


2.7760 


9329 


.70 


.7586 


1.2552 


.6044 


1.70 


2.6456 


2.8283 


.9354 


72 


7838 


1.2706 


6169 


1.72 


2.7027 


2.8818 


9379 


74 


8094 


1.2865 


6291 


1.74 


2.7609 


2.9364 


9402 


76 


8353 


1.3030 


6411 


1.76 


2.8202 


2.9922 


9425 


78 


8615 


1.3199 


6527 


1.78 


2.8806 


3.0492 


9447 


.80 


.8881 


1.3374 


.6640 


1.80 


2.9422 


3.1075 


.9468 


82 


9150 


1.3555 


6751 


1.82 


3.0049 


3.1669 


9488 


84 


9423 


1.3740 


6858 


1,84 


3.0689 


3.2277 


9508 


86 


9700 


1.3932 


6963 


1.86 


3.1340 


3.2897 


9527 


88 


9981 


1.4128 


7064 


1.88 


3.2005 


3.3530 


9545 


.90 


1.0265 


1.4331 


.7163 


1.90 


3.2682 


3.4177 


.9562 


92 


1.0554 


1.4539 


7259 


1.92 


3.3372 


3.4838 


9579 


94 


1.0847 


1.4753 


7352 


1.94 


3.4075 


3.5512 


9595 


96 


1.1144 


1.4973 


7443 


1.96 


3.4792 


3.6201 


9611 


98 


1.1446 


1.5199 


7531 


1.98 


3.5523 


3.6904 


9626 



TABLES. 



65' 



TABLE I. HYPERBOLIC FUNCTIONS. 



u. 


sinh u. 


cosh u. 


tanh . 


u. 


sinh u. 


cosh w. 


tanh u. 


2.00 


3.6269 


3.7622 


.9640 


3.00 


10.0179 


10.0677 


.99505 


2.02 


3.7028 


3.8355 


9654 


3.02 


10.2212 


10.2700 


99524 


2.04 


3.7803 


3.9103 


9667 


3.04 


10.4287 


10.4765 


99543 


2.06 


3.8593 


3.9867 


96bO 


3.06 


10.6403 


10.6872 


99561 


2.08 


3.9398 


4.0647 


9693 


3.08 


10.8562 


10.9022 


99578 


2.10 


4.0219 


4.1443 


.9705 


3.10 


11.0765 


11.1215 


.99594 


2.12 


4.1056 


4.2256 


9716 


3.12 


11.3011 


11.3453 


9961$ 


214 


4.1909 


4.3085 


9727 


3.14 


11.5303 


11.5786 


99626 


2.16 


42779 


4.3932 


9737 


3.16 


11.7641 


11.8065 


99640 


2.18 


4.3666 


4.4797 


9748 


3.18 


12.0026 


12.0442 


99654 


2.20 


4.4571 


4.5679 


.9757 


3.20 


12.2459 


12.2866 


.99668 


2.22 


4.5494 


4.6580 


9767 


3.22 


12.4941 


12.5340 


99681 


2.24 


4.6434 


4.7499 


9776 


3.24 


12.7473 


32.7864 


99693 


2.26 


4.7394 


4.8437 


9785 


3.26 


13 0056 


13.0440 


99705 


2.28 


4.8372 


4.9395 


- 9793 


3.28 


13.2691 


13.3067 


99717 


2.30 


4.9370 


5.0372 


.9801 


3.30 


13.5379 


13.5748 


.99728 


2.32 


5.0387 


5.1370 


9809 


332 


13.8121 


13.8483 


99738 


2.34 


5.1425 


5.2388 


9816 


3.34 


14.0918 


14.1273 


99749 


2.36 


5.2483 


5.3427 


9823 


3.36 


14.3772 


14.4120 


99758 


2.38 


5.3562 


5.4487 


9830 


3.38 


14.6684 


14.7024 


99768 


2.40 


5.4662 


5.5569 


.9837 


3.40 


14.9654 


14.9987 


.99777 


2.42 


5.5785 


5.6674 


9843 


3.42 


15.2684 


15.3011 


99786 


2.44 


5.6929 


5.7801 


9849 


3.44 


15.5774 


15.6095 


99794 


2.46 


58097 


5.8951 


9855 


3.46 


15.8928 


15.9242 


99802 


2.48 


5.9288 


6.0125 


9861 


3.48 


16.2144 


16.2453 


99810 


2.50 


6 0502 


6.1323 


.9866 


3.50 


16.5426 


16.5728 


.99817 


2 52 


6.1741 


6.2545 


9871 


3.52 


16.8774 


16.9070 


99824 


2.54 


6.3004 


6.3793 


9876 


3.54 


17.2190 


17.2480 


99831 


2.56 


6.4293 


6.5066 


9881 


3.56 


17.5674 


17.5958 


99838 


2.58 


6.5607 


6.6364 


9886 


3.58 


17.9228 


17.9507 


99844 


2.60 


6.6947 


6.7690 


.9890 


3.60 


18.2854 


18.3128 


.99850 


2.62 


6.8315 


6.9043 


9895 


3.62 


18.6554 


18.6822 


99856 


2.64 


6.9709 


7.04-23 


9899 


3.64 


19.0328 


19.0590 


99862 


2.66 


7.1132 


7.1832 


9903 


3.66 


19.4178 


19.4435 


99867 


2.68 


7.2583 


7.3268 


9906 


3.68 


19.8106 


19.8358 


99872 


2.70 


7.4063 


7.4735 


.9910 


3.70 


20.2113 


20.2360 


.99877 


2.72 


7.5572 


7.6231 


9914 


3.72 


20.6201 


20.6443 


99882 


2.74 


7.7112 


7.7758 


9917 


3.74 


21.0371 


21.0609 


99887 


2.76 


7.8683 


7.9316 


9920 


3.76 


21.4626 


21.4859 


99891 


2.78 


8.0285 


8.0905 


9923 


3.78 


21.8966 


21.9194 


99896 


2.80 


8.1919 


8.2527 


.9926 


3.80 


22.3394 


22.3618 


.99900 


2.82 


8.3586 


8.4182 


9929 


3.82 


22.7911 


22.8131 


99904 


2.84 


8.5287 


8.5871 


9932 


3.84 


23.2520 


23.2735 


99907 


2.86 


8.7021 


8.7594 


9935 


3.86 


23.7221 


23.7432 


99911 


2.88 


8.8791 


8.9352 


9937 


3.88 


24.2018 


24.2224 


99915 


2.90 


90596 


9.1146 


.9940 


3.90 


24.6911 


24.7113 


.9991 & 


2.92 


9.2437 


9.2976 


9942 


3.92 


25 1903 


25.2101 


99921 


2.94 


9.4315 


94844 


9944 


3.94 


25.6996 


25.7190 


99924 


2.96 


9.6231 


96749 


9947 


3.96 


26 2191 


26.2382 


99927 


2.98 


98185 


9.8693 


9949 


3.98 


26.7492 


26.7679 


99930 



66 HYPERBOLIC FUNCTIONS. 

TABLE II. VALUES OF COSH (x -f- iy) AND SINK (x -j- iy). 





X = O 


X = .1 


y 




















a 


b 


c 


d 


a 


b 


c 


d 





1.0000 


0000 


0000 


.0000 


1.0050 


.00000 


.10017 


.0000 


.1 


0.9950 


" 





0998 


1.0000 


01000 


09967 


1003 


.2 


0.9801 


< i 





1987 


0.9850 


01990 


09817 


1997 


.3 


0.9553 


i < 





2955 


0.9601 


02960 


09570 


2970 


.4 


.9211 


<( 





.3894 


.9257 


.03901 


.09226 


.3914 


.5 


8776 








4794 


8820 


04802 


08791 


4818 


.6 


8253 


< < 





5646 


8295 


05656 


08267 


5675 


.7 


7648 





K 


6442 


7687 


06453 


07661 


0474 


.8 


.6967 








.7174 


.7002 


.07186 


.06979 


.7200 


.9 


6216 








7833 


6247 


07847 


06227 


7872 


1.0 


5403 








8415 


5430 


08429 


05412 


8457 


1.1 


4536 


it 


ii 


8912 


4559 


08927 


04544 


8957 


1.2 


.3624 





< 


.9320 


.3642 


.09336 


.03630 


9367 


1.8 


2675 








9636 


2688 


09652 


02680 


0.9684 


1.4 


1700 


it 





9854 


1708 


09871 


01703 


0.9904 


1.5 


0707 





< < 


9975 


0711 


09992 


00709 


1.0025 


\7t 


0000 


< 





1.0000 


0000 


10017 


00000 


1.0050 


1/ 


x = .4 


x = .5 


y 


a 


b 


c 


d 


a 


b 


c 


d 





1.0811 


.0000 


.4108 


.0000 


1.1276 


.0000 


.5211 


.0000 


.1 


1.0756 


0410 


4087 


1079 


1 . 1220 


0520 


5185 


1126 


.2 


1.0595 


0816 


4026 


2148 


1.1051 


1025 


5107 


2240 


.3 


1.0328 


1214 


3924 


3195 


1.0773 


1540 


4978 


3332 


.4 


.9957 


.1600 


.3783 


.4210 


1.0386 


.2029 


.4800 


.4391 


.5 


9487 


1969 


3605 


5183 


0.9896 


2498 


4573 


5406 


.6 


8922 


2319 


3390 


6104 


0.9306 


2942 


4301 


6367 


.7 


8268 


2646 


3142 


6964 


0.8624 


3357 


3986 


7264 


.8 


.7532 


.2947 


.2862 


.7755 


.7856 


.3738 


.3631 


0.8089 


.9 


6720 


3218 


2553 


8468 


7009 


4082 


3239 


0.8833 


1.0 


5841 


3456 


2219 


9097 


6093 


4385 


2815 


0.9489 


1.1 


4904 


3661 


1863 


9635 


5115 


4644 


2364 


1.0050 


1.2 


.3917 


.3829 


.1488 


1.0076 


.4086 


.4857 


.1888 


1.0510 


1.3 


2892 


3958 


1099 


1.0417 


3016 


5021 


1394 


1.0865 


1.4 


1838 


4048 


0698 


1.0653 


1917 


5135 


0886 


1.1163 


1.5 


0765 


4097 


0291 


1.0784 


0798 


5198 


0369 


1 . 1248 


\Tt 


0000 


4108 


0000 


1.0811 


0000 


5211 


0000 


1.1276 



TABLES. 

TABLE II. VALUES OF COSH (x -f- iy) AND SINH (.* -f- iy\ 



X = .2. 


x = .3 




















y 


a 


b 


c 


d 


a 


b 


c 


d 




1.0201 


.0000 


.2013 


.0000 


1.0453 


.0000 


.3045 


.0000 





1.0150 


0201 


2003 


1018 


1.0401 


0304 


3030 


1044 


.1 


0.9997 


0400 


1973 


2027 


1.0245 


0605 


2985 


2077 


.2 


0.9745 


0595 


1923 


3014 


9987 


0900 


2909 


3089 


.3 


.9395 


.0784 


.1854 


.3972 


.9628 


.1186 


.2805 


.4071 


.4 


8952 


0965 


1767 


4890 


9174 


1460 


2672 


5012 


.5 


8419 


1137 


1662 


5760 


8627 


1719 


2513 


5903 


.6 


7802 


1297 


1540 


8571 


7995 


1962 


2329 


6734 


.7 


.7107 


.1444 


.1403 


.7318 


.7283 


.2184 


.2122 


.7498 


.8 


6341 


1577 


1252 


7990 


6498 


2385 


1893 


8188 


.9 


5511 


1694 


1088 


8584 


5648 


2562 


1645 


8796 


1,0 


4627 


1795 


0913 


9091 


4742 


2714 


1381 


9316 


1.1 


.3696 


.1877 


.0730 


0.9507 


.3788 


.2838 


.1103 


0.9743 


1.2 


2729 


1940 


0539 


0.9829 


2796 


2934 


0815 


1.0072 


1.8 


1734 


1984 


0342 


1.0052 


1777 


3001 


0518 


1.0301 


1.4 


0722 


2008 


0142 


1.0175 


0739 


3038 


0215 


1.0427 


1.5 


0000 


2013 


0000 


1.0201 


0000 


3045 


0000 


1.0453 


\K 


x = .6 


x = .7 




















y 


a 


b 


c 


d 


a 


b 


c 


d 




1.1855 


.0000 


.6367 


.0000 


1.2552 


.0000 


.7586 


.0000 





1.1795 


0636 


6335 


1183 


1.2489 


0757 


7548 


1253 


.1 


1.1618 


1265 


6240 


2355 


1.2301 


1542 


7435 


2494 


.2 


1 . 1325 


1881 


6082 


3503 


1.1991 


2242 


7247 


3709 


.3 


1.0918 


.2479 


.5864 


.4617 


1.1561 


.2954 


.6987 


.4888 


.4 


1.0403 


3052 


5587 


5684 


1.1015 


3637 


6657 


6018 


.5 


0.9784 


3595 


5255 


6694 


1.0359 


4253 


6261 


7087 


.6 


0.9067 


4101 


4869 


7637 


0.9600 


4887 


5802 


8086 


.7 


.8259 


.4567 


.4436 


0.8504 


.8745 


.5442 


.5285 


0.9004 


.8 


7369 


4987 


3957 


0.9286 


7802 


5942 


4715 


0.9832 


.9 


6405 


5357 


3440 


0.9975 


6782 


6383 


4099 


1.0562 


1.0 


5377 


5674 


2888 


1.0565 


5693 


6760 


3441 


1.1186 


1.1 


.4296 


5934 


.2307 


1.1049 


.4548 


.7070 


.2749 


1.1699 


1.2 


3171 


6135 


1703 


1.1422 


3358 


7309 


2029 


1.2094 


1.3 


2015 


6374 


1082 


1,1682 


2133 


7475 


1289 


1.2369 


1.4 


0839 


6351 


0450 


1.18.25 


0888 


7567 


0537 


1.2520 


1.5 


0000 


6367 


0000 


1.1855 


0000 


7586 


0000 


1.2552 


\1t 



68 HYPERBOLIC FUNCTIONS. 

TABLE II. VALUES OF COSH (x -f- iy) AND SINH(.# + iy). 





x = .8 


x = .9 


y 


s 


a 


b 


c 


d 


a 


b 


c 


d 





1.3374 


.0000 


.8881 


.0000 


1.4331 


.0000 


1.0265 


.0000 


.1 


1 . 3308 


0887 


8837 


1335 


1.4259 


1025 


1.0214 


1431 


.2 


1.3108 


1764 


8704 


2657 


1.4045 


2039 


1.0061 


2847 


.3 


1.2776 


2625 


8484 


3952 


1.3691 


3034 


0.9807 


4235 


A 


1.2319 


.3458 


.8180 


.5208 


1.3200 


.3997 


.9455 


.5581 


.5 


1.1737 


4258 


7794 


6412 


1.2577 


4921 


9008 


6871 


.6 


1.1038 


5015 


7330 


7552 


1.1828 


5796 


8472 


8092 


.7 


1.0229 


5721 


6793 


8616 


1.0961 


6613 


7851 


9232 


.8 


.9318 


.6371 


.6188 


0.9595 


.9984 


.7364 


.7152 


1.0280 


.9 


8314 


6957 


5521 


1.0476 


8908 


8041 


6381 


1.1226 


1.0 


7226 


7472 


4798 


1.1254 


7743 


8638 


5546 


1.2059 


1.1 


6067 


7915 


4028 


1.1919 


6500 


9148 


4656 


1.2772 


1.2 


.4846 


.8278 


.3218 


1.2465 


.5193 


0.9568 


.3720 


1.3357 


1.3 


3578 


8557 


2376 


1.2887 


3834 


0.9891 


2746 


1.3809 


1.4 


2273 


8752 


1510 


1.3180 


2436 


1.0124 


1745 


1.4122 


1.5 


0946 


8859 


0628 


1.3341 


1014 


1.0239 


0726 


1.4295 


i* 


0000 


.8881 


0000 


1.3374 


0000 


1.0265 


0000 


1.4331 




X = 1.2 


x = 1.3 


y 


a 


b 


c 


d 


a b 


c 


d 





1.8107 


.0000 


1.5095 


.0000 


1.9709 


.0000 


1.6984 


.0000 


.1 


1.8016 


1507 


1.5019 


1808 


1.9611 


1696 


1.6899 


1968 


.2 


1.7746 


2999 


1.4794 


3598 


1.9316 


3374 


1.6645 


3916 


.3 


1.7298 


4461 


1.4420 


5351 


1.8829 


5019 


1.6225 


5824 


.4 


1.6677 


.5878 


1.3903 


0.7051 


1.8153 


.6614 


1.5643 


0.7675 


.5 


1.5890 


7237 


1.3247 


0.8681 


1.7296 


8142 


1.4905 


0.9449 


.6 


1.4944 


8523 


1.2458 


1.0224 


1 6267 


9590 


1.4017 


1.1131 


.7 


1.3849 


9724 


1.1545 


1.1665 


1.5074 


1.0941 


1.2990 


1.2697 


.8 


1.2615 


1.0828 


1.0517 


1.2989 


1.3731 


1.2183 


1.1833 


1.4139 


.9 


1.1255 


1.1824 


0.9383 


1.4183 


1 2251 


1.3304 


1.0557 


1.5439 


1.0 


0.9783 


1.2702 


0.8156 


1.5236 


1.0649 


1.4291 


0.9176 


1.6585 


1.1 


0.8213 


1.3452 


0.6847 


1.6137 


0.8940 


1.5136 


0.7704 


1.7565 


1.2 


.6561 


1.4069 


.5470 


1.6876 


.7142 


1.5830 


.6154 


1.8370 


1.3 


4844 


1.4544 


4038 


1.7447 


5272 


1.6365 


4543 


1.8991 


1.4 


3078 


1.4875 


2566 


1.7843 


3350 


1.6737 


2887 


1.9422 


1.5 


1281 


1.5057 


1068 


1.8061 


1394 


1.6941 


1201 


1.9660 


*r 


0000 


1.5095 


0000 


1.8107 


0000 


1.6984 


0000 


1.9709 



TABLES. 69 

TABLE II. VALUES OF COSH (x -\- iy) AND SINK (x -f- iy.) 



X = I.O 


X = I.I 


I 

y 


a b 


c d 


a b 


c d 


1.5431 .0000 
1.5354 1173 
1.5123 2335 
1.4742 3473 


1.1752 .0000 
1.1693 1541 
1.1518 3066 
1.1227 4560 


1.6685 .0000 
1.6602 1333 
1.6353 2654 
1.5940 3946 


1.3356 .0000 
1.3290 1666 
1.3090 3315 
1.2760 4931 




.1 

.2 
.3 


1.4213 .4576 
1.3542 5634 
1.2736 6636 
1.1802 7571 


1.0824 .6009 
1.0314 7398 
0.9699 8718 
0.8988 9941 


1.5368 .5201 
1.4643 6403 
1.3771 7542 
1.2762 8604 


1.2302 0.6498 
1.1721 0.7999 
1.1024 0.9421 
1.0216 1,0749 


.4 
.5 
.6 

.7 


1.0751 0.8430 
0.9592 0.9206 
0.8337 0.9889 
0.6999 1.0473 


.8188 1 1069 
7305 1.2087 
6350 1.2985 
5331 1.3752 


1.1625 0.9581 
1.0372 1.0462 
0.9015 1.1239 
0.7568 1.1903 


.9306 1.1969 
8302 1.3070 
7217 1.4040 
6058 1.4870 


.8 
.9 
1.0 

i.l 


.5592 1.0953 
4128 1.1324 
2623 1.1581 
1092 1.1723 


.4258 1.4382 
3144 1.4869 
1998 1.5213 
0831 1.5392 


.6046 1.2449 
4463 1.2870 
2836 1.3162 
1180 1.3323 


.4840 1.5551 
3573 1.6077 
2270 1.6442 
0945 1.6643 


1.2 
1.3 
1.4 
1.5 


0000 1.1752 


0000 1.5431 


0000 1.3356 


0000 1.6685 


** 

y 



.1 

.2 
.3 


x = 1.4 


x - 1.5. 


a b 


c d 


a b 


c d 


2.1509 .0000 
2.1401 1901 
2.1080 3783 
2.0548 5628 


1.9043 .0000 
1.8948 2147 
1.8663 4273 
1.8192 6356 


2.3524 .0000 
23413 2126 
2.3055 4230 
2.2473 6292 


2.1293 .0000 
2.1187 2348 
2.0868 4674 
2.0342 6951 


1.9811 0.7416 
1.8876 0.9130 
1.7752 1.0753 
1.6451 1.2268 


1.7540 0.8376 
1.6712 1.0312 
1.5713 1.2145 
1.4565 1.3856 


2.1667 0.8292 
2.0644 1.0208 
1.9415 1.2023 
1.7992 1.3717 


1.9612 0.9161 
1.8686 1.1278 
1.7574 1.3283 
1.6286 1.5155 


.4 
.5 
.6 

.7 


1.4985 1.3661 
1.3370 1.4917 
1.1622 1.6024 
0.9756 1.6971 


1.3268 1.5430 
1.1838 1.6849 
1.0289 1.8099 
0.8638 1.9168 


1.6389 1.5275 
1.4623 1.6679 
1.2710 1.7917 
1.0671 1.8976 


1.4835 1.6875 
1.3236 1.8427 
1.1505 1.9795 
0.9659 2.0965 


.8 
.9 
1.0 
1.1 


.7794 1.7749 
5754 1.8349 
3656 1.8766 
1522 1.8996 


.6900 2.0047 
5094 2.0725 
3237 2.1196 
1347 2.1455 


.8524 1.9846 
6293 2.0517 
3998 2.0983 
1664 2.1239 


.7716 2.1925 
5696 2.2667 
3619 2.3182 
1506 2.3465 


1.2 
1.3 
1.4 
1.5 


.0000 1.9043 


0000 2.1509 


.0000 2.1293 


.0000 2.3524 


\n 



70 



HYPERBOLIC FUNCTIONS. 



TABLE III. 



u 


gdu Q 


u 


gd u B 


u 


gd u Q 







o 




o 


00 


.0000 0.000 


.60 


.5669 32.483 


1.50 


1.1317 64.843 


.02 


0200 1 . 146 


.62 


5837 33.444 


1.55 


1.1525 66.034 


.04 


0400 2.291 


.64 


6003 34.395 


1.60 


1.1724 67.171 


.06 


0600 3.486 


.66 


6167 35 336 


1.65 


1.1913 68.257 


.08 


0799 4.579 


.68 


63:29 36.265 


1.70 


1.2094 69.294 


.10 


.0998 5.720 


.70 


.6489 37.183 


1.75 


1.2267 70.284 


.12 


1197 6.859 


.72 


6648 38.091 


1.80 


1.2432 71.228 


.14 


1395 7.995 


.74 


6804 38.987 


1.85 


1.2589 72.128 


.16 


1593 9.128 


.76 


6958 39.872 


1.90 


1.2739 72.987 


.18 


1790 10.258 


.78 


7111 40.746 


1.95 


1.2881 73.805 


.20 


.1987 11.384 


.80 


.7261 41.608 


2.00 


1.3017 74.584 


.23 


2183 12.505 


.82 


7410 42.460 


2.10 


1.3271 76.037 


.24 


2377 13.621 


.84 


7557 43.299 


2.20 


1.3501 77.354 


.26 


2571 14.732 


.86 


7702 44.128 


2.30 


1.3710 78.549 


.28 


2764 15.837 


.88 


7844 44.944 


2.40 


1.3899 79.633 


.30 


.2956 16.937 


.90 


.7985 45.750 


2.50 


1.4070 80.615 


.32 


3147 18.030 


.92 


8123 46.544 


2.60 


1.4327 81.513 


.34 


3336 19.116 


.94 


8260 47.326 


2.70 


1.4366 82.310 


.36 


3525 *80.195 


.96 


8394 48.097 


2.80 


1.4493 83.040 


.38 


3712 21.267 


.98 


8528 48.857 


2.90 


1.4609 83.707 


.40 


.3897 22.331 


1.00 


.8658 49.605 


300 


1.4713 84.301 


.42 


4082 23.386 


1.05 


8976 51.428 


3.10 


1.4808 84.841 


.44 


4264 24.434 


1.10 


9581 53.178 


3.20 


1.4894 85.33(> 


.46 


4446 25.473 


1.15 


9575 54 860 


3.30 


1.4971 85. 7; 5 


.48 


4626 26.503 


1.20 


9857 56.476 


3.40 


1.5041 86.177 


.50 


.4804 27.524 


1.25 


1.0127 58.026 


3.50 


1.5104 86.541 


.52 


4980 28.535 


1.30 


1.0387 59.511 


3.60 


1.5162 86.870 


.54 


5155 29.537 


1.35 


1.0635 60.933 


3.70 


1.5214 87 16s 


.56 


5328 30.529 


1.40 


1.0873 62.295 


3.80 


1.5261 87.437 


.58 5500 31.511 


1.45 


1.1100 63.598 


3.90 


1.5303 87.681 



TABLE IV. 



u 


gd u log sinh u 


log cosh u 


u 


gd u 


log sinh u 


log cosh u 


4.0 


1.5342 1.4360 


1.4363 


5.5 


1.5626 


2.08758 


2.08760 


4.1 


1.5377 


1.4795 


1.4797 


5.6 


1.5634 


2.13101 


2.13103 


4.2 


1.5408 


1.5229 


1.5231 


5.7 


1.5641 


2.17444 


2.17445 


4.3 


1 5437 


1.5664 


1.5665 


5.8 


1.5648 


2.21787 


2.21788 


4.4 


1.5462 


1.6098 


1.6099 


5.9 


1.5653 


2.26130 


2.26131 


4.5 


1.5486 


1.6532 


1.6533 


60 


1.5658 


2.30473 


2.30474 


4.6 


1.5507 


1 6967 


1.6968 


6.2 


1.6867 


2.39159 


8.89160 


4.7 


1 5526 


1.7401 


1.7402 


6.4 


1.5675 


2.47845 


2.47H-H) 


4.8 


1.5543 


1.7836 


1.7836 


6.6 


1.5681 


2.56531 


2.56531 


4.9 


1.5559 


1.8270 


1.8270 


6.8 


1.5686 


2.65217 


2.65217 


5.0 


1 5573 


1.8704 


1.8705 


7.0 


1.5690 


2.73903 


2.73903 


5.1 


1.5586 


1.9139 


1.9139 


7.5 


1.5697 


2.95618 


3. 956 is 


5.2 


1.5598 


1.9573 


1.9573 


8.0 


1.5701 


3.17333 


8.17333 


5.3 


1.5608 


2.0007 


2.0007 


8.5 


1.5704 


3.39047 


3.39047 


5.4 


1 5618 


2.0442 


2.0442 


9.0 


1.5705 


3.60762 


3.60762 










GO 


1.5708 


GO 


00 



APPENDIX. 



HISTORICAL AND BIBLIOGRAPHICAL. 

is probably the earliest suggestion of the analogy between 
the sector of the circle and that of the hyperbola is found in Newton's 
Principia (Bk. 2, prop. 8 et seq.) in connection with the solution of a 
dynamical problem. On the analytical side, the first hint of the modi- 
fied sine and cosine is seen in Roger Cotes' Harmonica Mensurarum 
(1722), where he suggests the possibility of modifying the expression 
for the area of the prolate spheroid so as to give that of the oblate one, 
by a certain use of the operator V i. The actual inventor of the 
hyperbolic trigonometry was Vincenzo Riccati, S.J. (Opuscula ad res 
Phys. et Math, pertinens, Bononise, 1757). He adopted the notation 
Sh.^>, Ch.< for the hyperbolic functions, and Sc.^>, Cc.< for the cir- 
cular ones. He proved the addition theorem geometrically and derived 
a construction for the solution of a cubic equation. Soon after, Daviet 
de Foncenex showed how to interchange circular and hyperbolic func- 
tions by the use of V i, and gave the analogue of De Moivre's theorem, 
the work resting more on analogy, however, than on clear definition 
(Reflex, sur les quant, imag., Miscel. Turin Soc., Tom. i). Johann 
Heinrich Lambert systematized the subject, and gave the serial devel- 
opments and the exponential expressions. He adopted the notation 
sinh u, etc., and introduced the transcendent angle, now called the 
gudermanian, using it in computation and in the construction of tables 
(1. c. page 30). The important place occupied by Gudermann in the 
history of the subject is indicated on page 30. 

The analogy of the circular and hyperbolic trigonometry naturally 
played a considerable part in the controversy regarding the doctrine 
of imaginaries, which occupied so much attention in the eighteenth cen- 
tury, and which gave birth to the modern theory of functions of the 



72 HYPERBOLIC FUNCTIONS. 

complex variable. In the growth of the general complex theory, the 
importance of the " singly periodic functions" became still clearer, and 
was gradually developed by such writers as Ferroni (Magnit. expon. 
log. et trig., Florence, 1782); Dirksen (Organon der tran. Anal., Ber- 
lin, 1845); Schellbach (Die einfach. period, funkt., Crelle, 1854); Ohm 
(Versuch eines volk. conseq. Syst. der Math., Niirnberg, 1855); Hotiel 
(Theor. des quant, complex, Paris, 1870). Many other writers have 
helped in systematizing and tabulating these functions, and in adapting 
them to a variety of applications. The following works may be espe- 
cially mentioned: Gronau (Tafeln, 1862, Theor. und Anwend., 1865); 
Forti (Tavoli e teoria, 1870); Laisant (Essai, 1874); Gunther (Die 
Lehre . . . , 1881). The last-named work contains a very full history 
and bibliography with numerous applications. Professor A. G. Green- 
hill, in various places in his writings, has shown the importance of both 
the direct and inverse hyperbolic functions, and has done much to pop- 
ularize their use (see Diff. and Int. Calc., 1891). The following articles 
on fundamental conceptions should be noticed: Macfarlane, On the 
definitions of the trigonometric functions (Papers on Space Analysis, 
N. Y., 1894); Haskell, On the introduction of the notion of hyperbolic 
functions (Bull. N. Y. M. Soc., 1895). Attention has been called in 
Arts. 30 and 37 to the work of Arthur E. Kennelly in applying the 
hyperbolic complex theory to the plane vectors which present them- 
selves in the theory of alternating currents; and his chart has been 
described on page 44 as a useful substitute for a numerical complex 
table (Proc. A. I. E. E., 1895). It may be worth mentioning in this 
connection that the present writer's complex table in Art. 39 is believed 
.to be the only one of its kind for any function of the general argument 
x+iy. 



EXPONENTIAL EXPRESSIONS AS DEFINITIONS. 

For those who wish to start with the exponential expressions as the 
definitions of sinh u and cosh u, as indicated on page 25, it is here pro- 
posed to show how these definitions can be easily brought into direct 
geometrical relation with the hyperbolic sector in the form #/# = cosh 
S/K, y/b = sinh S/K, by making use of the identity cosh 2 u sinh 2 u= i, 
and the differential relations d cosh w=sinh u du, d sinh u= cosh u du, 
which are themselves immediate consequences of those exponential 
definitions. Let OA, the initial radius of the hyperbolic sector, be 



EXPONENTIAL EXPRESSIONS AS DEFINITIONS. 73 

taken as axis of x, and its conjugate radius OB as axis of y\ let OA = a, 
= b, angle AOB = w, and area of triangle AOB=K, then K= 
sin aj. Let the coordinates of a point P on the hyperbola be x 
.and ;y, then x 2 /a 2 y 2 /b 2 =i. Comparison of this equation with the 
identity cosh 2 u sinh 2 u=i permits the two assumptions x/a=coshu 
and ;y/6=sinh u, wherein u is a single auxiliary variable; and it now 
remains to give a geometrical interpretation to u, and to prove that 
u=S/K, wherein 6" is the area of the sector OAP. Let the coordinates 
of a second point Q be x+ Ax and y+dy, then the area of the triangle 
POQ is, by analytic geometry, %(xdy ydx)sm to. Now the sector 
POQ bears to the triangle POQ a ratio whose limit is unity, hence the 
differential of the sector 5 may be written dS=%(x dy y dx)sm a>= 
\ab sin at (cosh 2 u sinh 2 u)du=K du. By integration S=Ku, hence 
u=S/K, the sectorial measure (p. 10); this establishes the fundamental 
geometrical relations x/a=cosh S/K, y/b=siuh S/K. 



INDEX. 



Addition-theorems, pages 16, 40. 
Admittance of dielectric, 56. 
Algebraic identity, 41. 
Alternating currents, 38, 46, 55. 
Ambiguity of value, 13, 16, 45. 
Amplitude, hyperbolic, 31. 

of complex number, 46. 
Anti-gudermanian, 28, 30, 47, 51, 52. 
Anti-hyperbolic functions, 16, 22, 25, 29, 

35. 45- 
Applications, 46 et seq. 

Arch, 48, 51. 

Areas, 8, 9, 14, 36, 37, 60. 

Argand diagram, 43, 58. 

Bassett's Hydrodynamics, 61. 
Beams, flexure of, 54. 
Bedell and Crehore's alternating cur- 
rents, 38, 56. 
Byerly's Fourier Series, etc., 61, 63. 

Callet's Tables, 63. 
Capacity of conductor, 55. 
Catenary, 47. 

of uniform strength, 49. 
Elastic, 48. 

Cay ley's Elliptic Functions, 30, 31. 
Center of gravity, 61. 
Characteristic ratios, 10. 
Chart of hyperbolic functions, 44, 58. 

Mercator's, 53. 
Circular functions, 7, n, 14, 18, 21, 24, 

29, 35. 4i, 43- 

of complex numbers, 39, 41, 42. 
of gudermanian, 28. 



Complementary triangles, 10. 
Complex numbers, 38-46. 

Applications of, 55-60. 

Tables, 62, 66. 
Conductor resistance and impedance, 

58. 

Construction for gudermanian, 30. 
of charts, 43. 
of graphs, 32. 
Convergence, 23, 25. 
Conversion-formulas, 18. 
Corresponding points on conies, 7, 28. 

sectors and triangles, 9, 28. 
Currents, alternating, 55. 
Curvature, 50, 52, 60. 
Cotes, reference to, 71. 

Deflection of beams, 54. 

Derived functions, 20, 22, 30. 

Difference formula, 16. 

Differential equation, 21, 25, 47, 49, 51, 

52, 57- 

Dirksen's Organon, 71. 
Distributed load, 55. 

Electromotive force, 55, 58. 
Elimination of constants, 21. 
Ellipses, chart of confocal, 43. 
Elliptic functions, 7, 30, 31. 

integrals, 7, 31. 

sectors, 7, 31. 
Equations, Differential (see). 

Numerical, 35, 48, 50. 
E volute of tractory, 52. 
Expansion in series, 23, 25, 31. 



76 



INDEX. 



Exponential expressions, 24, 25, 72. 

Ferroni, reference to, 71. 
Flexure of beams, 53. 
Foncenex, reference to, 71. 
Ford's Tavoli e teoria, 63, 71. 
Fourier series, 55, 61. 
Function, anti-gudermanian (see). 

anti-hyperbolic (see). 

circular (see). 

elliptic (see). 

gudermanian (see). 

hyperbolic, denned, n. 

of complex numbers, 38. 

of pure imaginaries, 41. 

of sum and difference, 16. 

periodic, 44. 

Geipel and Kilgour's Electrical Hand- 
book, 63. 

Generalization, 41. 
Geometrical interpretation, 37. 

treatment of hyperbolic 

functions, jetseq., 16. 
Glaisher's exponential tables, 63. 
Graphs, 32. 
Greenhill's Calculus, 72. 

Elliptic Functions, 7. 
Gronau's Tafeln, 63, 72. 

Theor. und Anwend., 72. 
Gudermann's notation, 30. 
Gudermanian, angle, 29. 

function, 28, 31, 34,47, 53, 63, 70. 
Gunther's Die Lehre, etc., 61, 71. 

Haskell on fundamental notions, 72. 
HoiiePs notation, etc., 30, 31, 71. 
Hyperbola, 7 et seq., 30, 37, 44, 60. 
Hyperbolic functions, denned, n. 

addition-theorems for, 16. 

applications of, 46 et seq. 

derivatives of, 20. 

expansions of, 23. 

exponential expressions for, 24. 

graphs of, 32. 

integrals involving, 35. 



Hyperbolic functions of complex num- 
bers, 38 et seq. 
relations among, 12. 
relations to gudermanian, 29. 
relations to circular functions, 29, 42. 
tables of, 64 et seq. 
variation of, 20. 

Imaginary, see complex. 

Impedance, 34. 

Integrals, 35. 

Interchange of hyperbolic and circular 

functions, 42. 

Interpolation, 30, 48, 50, 59, 62. 
Intrinsic equation, 38, 47, 49, 51. 
Involute of catenary, 48. 
of tractory, 50. 

Jones' Trigonometry, 52. 

Kennelly on alternating currents, 38, 58* 
Kennelly's chart, 46, 58. 

Laisant's Essai, etc., 61, 71. 
Lambert's notation, 30. 

place in the history, 70. 
Leakage of conductor, 55. 
Limiting ratios, 19, 23, 32. 
Logarithmic curve, 60. 

expressions, 27, 32. 
Love's elasticity, 61. 
Loxodrome, 52. 

Macfarlane on definitions, 72. 
Maxwell's Electricity, 61. 
Measure, defined, 8. 

of sector, 9 et seq* 
Mercator's chart, 53. 
Modulus, 31, 46. 
Moment of inertia, 61. 
Multiple values, 13, 16, 45. 

Newton, reference to, 71. 
Numbers, complex, 38 et seq. 

Ohm, reference to, 71. 
Operators, generalized, 39, 56. 

Parabola, 38, 61. 
Periodicity, 44, 62. 



INDEX. 



77- 



Permanence of equivalence, 41. 

Phase angle, 56, 59. 

Physical problems, 21, 38, 47 et seq. 

Potential theory, 6l. 

Product-series, 43. 

Pure imaginary, 41. 

Ratios, characteristic, 10. 

limiting, 19. 

Rayleigh's Theory of Sound, 61. 
Reactance of conductor, 58. 
Reduction formula, 37, 38. 
Relations among functions, 12, 29, 42. 
Resistance of conductor, 56. 
Rhumb line, 53. 
Riccati's place in the history, 71. 

Schellbach, reference to, 71. 
Sectors of conies, 9, 28. 



Self-induction of conductor, 55. 

Series, 23, 31. 

Spheroid, area of oblate, 58 

Spiral of Archimedes, 60. 

Steinmetz on alternating currents, 38,. 

55- 
Susceptance of dielectric, 58. 

Tables, 43, 62. 

Terminal conditions, 54, 58, 60. 

Tractory, 48, 51. 

Variation of hyperbolic functions, 14^ 
Vassal's Tables, 63. 
Vectors, 38, 56. 
Vibrations of bars, 61. 

Wheeler's Trigonometry, 61. 



342 
M27 
1906 



McMahon, James 

Hyperbolic functions 
4th ed., enl. 



Physical -fit 
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